Class 10th Mathematics AP Board Solution
Exercise 8.1- In Δ PQR, ST is a line such that ps/sq = pt/tr and also ∠PST = ∠PRQ. Prove that…
- In the given figure, LM || CB and LN || CD Prove that am/ab = an/ad left arrow…
- In the given figure, DE||AC and DF||AE Prove that bf/fe = be/ec
- In the given figure, AB||CD||EF. given AB = 7.5 cm, DC = ycm, EF = 4.5cm, BC = x…
- Prove that a line drawn through the mid-point of one side of a triangle parallel…
- Prove that a line joining the midpoints of any two sides of a triangle is…
- In the given figure, DE||OQ and DF||OR. Show that EF||QR.
- In the adjacent figure, A, B, and C are points on OP, OQ and Or respectively…
- ABCD is a trapezium in which AB||DC and its diagonals intersect each other at…
- Draw a line segment of length 7.2 cm and divide it in the ratio 5 : 3. Measure…
Exercise 8.2- In the given figure, ∠ADE = ∠B i. Show that ΔABC ~ ΔADE ii. If AD = 3.8cm, AE =…
- The perimeters of two similar triangles are 30 cm and 20 cm respectively. If one…
- A girl of height 90 cm is walking away from the base of a lamp post at a speed…
- Given that Δ ABC ∼ Δ PQR, CM and RN are respectively the medians of Δ ABC and Δ…
- Diagonals AC and BD of a trapezium ABCD with AB||DC intersect each other at the…
- AB, CD, PQ are perpendicular to BD. AB = x, CD = y and PQ = z prove that 1/x +…
- A flag pole 4 m tall casts a 6 m., shadow. At the same time, a nearby building…
- CD and GH are respectively the bisectors of ∠ACE and ∠EGF such that D and H lie…
- AX and DY are altitudes of two similar Δ ABC and ΔDEF. Prove that AX : DY = AB :…
- Construct a triangle shadow similar to the given ΔABC, with its sides equal to…
- Construct a triangle of sides 4 cm, 5 cm and 6 cm. Then, construct a triangle…
- Construct an Isosceles triangle whose base is 8 cm and altitude is 4 cm. Then,…
Exercise 8.3- Equilateral triangles are drawn on the three sides of a right angled triangle.…
- Prove that the area of the equilateral triangle described on the side of a…
- D, E, F are mid points of sides BC, CA, AB to Δ ABC. Find the ratio of areas of…
- In Δ ABC, XY || AC and XY divides the triangle into two parts of equal area.…
- Prove that the ratio of areas of two similar triangles is equal to the square of…
- Δ ABC ∼ Δ DEF. BC = 3 cm EF = 4 cm and area of ΔABC = 54cm^2 . Determine the…
- ABC is a triangle and PQ is a straight line meeting AB and P and AC in Q. If AP…
- The areas of two similar triangles are 81cm^2 and 49cm^2 respectively. If the…
Exercise 8.4- Prove that the sum of the squares of the sides of a rhombus is equal to the sum…
- ABC is a right triangle right angled at B. Let D and E be any points on AB and…
- Prove that three times the square of any side of an equilateral triangle is…
- PQR is a triangle right angled at P and M is a point on QR such that PM ⊥ QR.…
- ABD is a triangle right angled at A and AC ⊥ BD Show that i. AB^2 = BC . BD ii.…
- ABC is an isosceles triangle right angled at C. Prove that AB^2 = 2AC^2 .…
- ‘O’ is any point in the interior of a triangle ABC. OD ⊥ BC, OE ⊥ AC and OF ⊥…
- A wire attached to vertically pole of height 18m is24m long and has a stake…
- Two poles of heights 6m and 11m stand on a plane ground. If the distance between…
- In an equilateral triangle ABC, D is on a side BC such that bd = 1/3 bc Prove…
- In the given figure, ABC is a triangle right angled at B. D and E are points on…
- ABC is an isosceles triangle right angled at B. Similar triangles ACD and ABE…
- In Δ PQR, ST is a line such that ps/sq = pt/tr and also ∠PST = ∠PRQ. Prove that…
- In the given figure, LM || CB and LN || CD Prove that am/ab = an/ad left arrow…
- In the given figure, DE||AC and DF||AE Prove that bf/fe = be/ec
- In the given figure, AB||CD||EF. given AB = 7.5 cm, DC = ycm, EF = 4.5cm, BC = x…
- Prove that a line drawn through the mid-point of one side of a triangle parallel…
- Prove that a line joining the midpoints of any two sides of a triangle is…
- In the given figure, DE||OQ and DF||OR. Show that EF||QR.
- In the adjacent figure, A, B, and C are points on OP, OQ and Or respectively…
- ABCD is a trapezium in which AB||DC and its diagonals intersect each other at…
- Draw a line segment of length 7.2 cm and divide it in the ratio 5 : 3. Measure…
- In the given figure, ∠ADE = ∠B i. Show that ΔABC ~ ΔADE ii. If AD = 3.8cm, AE =…
- The perimeters of two similar triangles are 30 cm and 20 cm respectively. If one…
- A girl of height 90 cm is walking away from the base of a lamp post at a speed…
- Given that Δ ABC ∼ Δ PQR, CM and RN are respectively the medians of Δ ABC and Δ…
- Diagonals AC and BD of a trapezium ABCD with AB||DC intersect each other at the…
- AB, CD, PQ are perpendicular to BD. AB = x, CD = y and PQ = z prove that 1/x +…
- A flag pole 4 m tall casts a 6 m., shadow. At the same time, a nearby building…
- CD and GH are respectively the bisectors of ∠ACE and ∠EGF such that D and H lie…
- AX and DY are altitudes of two similar Δ ABC and ΔDEF. Prove that AX : DY = AB :…
- Construct a triangle shadow similar to the given ΔABC, with its sides equal to…
- Construct a triangle of sides 4 cm, 5 cm and 6 cm. Then, construct a triangle…
- Construct an Isosceles triangle whose base is 8 cm and altitude is 4 cm. Then,…
- Equilateral triangles are drawn on the three sides of a right angled triangle.…
- Prove that the area of the equilateral triangle described on the side of a…
- D, E, F are mid points of sides BC, CA, AB to Δ ABC. Find the ratio of areas of…
- In Δ ABC, XY || AC and XY divides the triangle into two parts of equal area.…
- Prove that the ratio of areas of two similar triangles is equal to the square of…
- Δ ABC ∼ Δ DEF. BC = 3 cm EF = 4 cm and area of ΔABC = 54cm^2 . Determine the…
- ABC is a triangle and PQ is a straight line meeting AB and P and AC in Q. If AP…
- The areas of two similar triangles are 81cm^2 and 49cm^2 respectively. If the…
- Prove that the sum of the squares of the sides of a rhombus is equal to the sum…
- ABC is a right triangle right angled at B. Let D and E be any points on AB and…
- Prove that three times the square of any side of an equilateral triangle is…
- PQR is a triangle right angled at P and M is a point on QR such that PM ⊥ QR.…
- ABD is a triangle right angled at A and AC ⊥ BD Show that i. AB^2 = BC . BD ii.…
- ABC is an isosceles triangle right angled at C. Prove that AB^2 = 2AC^2 .…
- ‘O’ is any point in the interior of a triangle ABC. OD ⊥ BC, OE ⊥ AC and OF ⊥…
- A wire attached to vertically pole of height 18m is24m long and has a stake…
- Two poles of heights 6m and 11m stand on a plane ground. If the distance between…
- In an equilateral triangle ABC, D is on a side BC such that bd = 1/3 bc Prove…
- In the given figure, ABC is a triangle right angled at B. D and E are points on…
- ABC is an isosceles triangle right angled at B. Similar triangles ACD and ABE…
Exercise 8.1
Question 1.In Δ PQR, ST is a line such that
and also ∠PST = ∠PRQ. Prove that ΔPQR, is an isosceles triangle.
![](data:image/png;base64,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)
Answer:Given, ![](data:image/png;base64,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)
Also, ∠PST = ∠PRQ
⇒ Need to prove that PQR is an isosceles triangle.
⇒ If a line divides any two sides of a triangles in the same ratio then the same line is parallel to the third side.
⇒ Since, ST || QR
⇒ ∠PST = ∠PQR …………eq(1)
(Corresponding angles)
Also given ∠PST = ∠PRQ ……………eq(2)
From eq(1) and eq(2) we get
⇒ ∠PQR = ∠PRQ
Since, sides opposite to equal angles are equal
⇒ PR = PQ
∴ two sides of
PQR is equal
PQR is an isosceles triangle
Hence proved.
Question 2.In the given figure, LM || CB and LN || CD Prove that ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Answer:Given, LM||CB and LN||CD
Need to prove ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEUAAAAeCAMAAABNGiMvAAAAAXNSR0IArs4c6QAAAJBQTFRFAAAAAAAAAAA6AABmADo6ADqQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOma2OpDbZgAAZgA6ZjqQZpBmZpDbZrb/kDoAkDo6kJA6kJBmkLaQkNuQkNv/tmYAtmY6tpA6ttv/tv+2tv//25A627Zm29uQ29v/2//b2////7Zm/7aQ/9uQ/9u2/9vb//+2///bMRPcZgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABXElEQVRIS+WU3VKDMBCFl9ZaqxZoldQfoNpUW0TJ+7+dZyMkYRyTXDidcdwLyAz7HTab7CH6tVDiHFpdnmR4yWS6CysbpGBO58srVvl6tZeTMixikdmRGk23yz3WRPX97Kg2dzGlGOTxoiRZAFaiaJjs1i+LncwqrewPixxEpjZcPLB2AZU27VYPy4Ng5UBYhJrp8xobaRIE90IWSpyVLUoMhYOgtzcp0Xt+y8dTEBdWZSR1i7wxQnCmKF4kaKnA6UossK25Pm5fINtFYooP1fVPv1d8DXT458KkOYtAy74jJ+lx7I5OUsxf/YlxUdwLjHhEOMT1U58/GC+meh9jmNaqMd11TwwuCntR2wjDtFYNQpsTwrhojlsZchddvSUg0KsYF4X7vkUYpmPV2q81Yl0Um+mF/f21BNtjxQ/rogJtGVrlkxkRHzVPu+uiaIs5tp9lxsQkfY24GFEpnyRUJGO855S6AAAAAElFTkSuQmCC)
⇒ In
ACB, LM || CB
⇒ now, we know that line drawn parallel to one side of the triangle, intersects the other two sides in distinct points, then it divides the other 2 side in same ratio.
∴
……….eq(1)
And in
ACD, LN||CD
⇒
…………eq(2)
From eq(1) and eq(2), we get
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEYAAAAeCAMAAACmLZgsAAAAAXNSR0IArs4c6QAAAJBQTFRFAAAAAAAAAAA6AABmADo6ADqQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOma2OpDbZgAAZgA6ZjqQZpBmZpDbZrb/kDoAkDo6kJA6kJBmkLaQkNuQkNv/tmYAtmY6tpA6ttv/tv+2tv//25A627Zm29uQ29v/2//b2////7Zm/7aQ/9uQ/9u2/9vb//+2///bMRPcZgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABcElEQVRIS+2VW0/DMAyFnQ0GY6wtg2Zc2gLNYCuF5v//O46TXlIJ1TxMk5DwQ+SqOSe2s30lOlI0iYphZdS8RKrU7E40xr6UukWp6xdWmNWCqL6aZUjPD7Sbl6KPWWFjBRkrbOEUxQPS7T3neUz2md8KUTxdZmRS7IICdSFrNm/L0sQ51K4p7lCIZrPXsd2ifO/ASx01N4/rveZ0WdIHzpGijqiav24OXkE1S0xq9VnmU1cR1zodaKdJbiM/GjSGhWtDg/xsNUbj5zUZrh2Dy3KKr8JdjmLD+gIzMZhMe3uTNk7iy+cfSPQunfv/fmICOU/dBV/HRPT7gkSY7A+S09zFr5s6TTl/9xSrmXgBlOVWekkLZVY4Go+gLPr0kg7KoMR6x7AIoSzZ9JIeymBPWjGrAihLLoOkgzIYuGiROkBZshkkHZSp6v54AZQFm0DSQfkzwVeSOR5AWXAZSTyUSQOsVuPLG0BZIPpY4qB8pPgGc1ooDkSO/rwAAAAASUVORK5CYII=)
Adding 1 on both sides we get
⇒
+ 1 =
+ 1
⇒
= ![](data:image/png;base64,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)
⇒
= ![](data:image/png;base64,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)
⇒
= ![](data:image/png;base64,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)
Hence, proved.
Question 3.In the given figure, DE||AC and DF||AE Prove that ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Answer:Given, DE || AC and DF || AE
Need to prove ![](data:image/png;base64,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)
⇒ In
ABC, DE || AC
⇒ now, we know line drawn parallel to one side of triangle, intersects the other two sides in distinct points, then it divides the other 2 side in same ratio.
⇒
………eq(1)
⇒ In
AEB, DF || AE
⇒
………eq(2)
From (1) and (2)
⇒ ![](data:image/png;base64,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)
Hence proved.
Question 4.In the given figure, AB||CD||EF. given
AB = 7.5 cm, DC = ycm, EF = 4.5cm, BC = x cm. Calculate the values of x and y.
![](data:image/png;base64,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)
Answer:Given, Ab = 7.5cm, DC = ycm, EF = 4.5cm and BC = xcm
Need to calculate values of x and y
⇒ Let us consider Δ ACB and Δ CEF
⇒ Both are similar triangles
∴ ![](data:image/png;base64,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)
⇒ x = 5cm
⇒ let us consider Δ BCD and Δ BFE
⇒ from basic proportionality theorem we have we know that line drawn parallel to one side of the triangle, intersects the other two sides in distinct points, then it divides the other 2 side in same ratio.
⇒ ![](data:image/png;base64,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)
⇒ y = ![](data:image/png;base64,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)
⇒ Y = ![](data:image/png;base64,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)
⇒ Y = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABwAAAAeCAMAAAAIG46tAAAAAXNSR0IArs4c6QAAAIdQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAtmY6tv//25A625Bm27Zm27aQ29u229vb2////7Zm/9uQ/9u2//+2///bPegm3wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAA4klEQVQ4T5VSWRbCIAwMLrV1q3tFLRWtdjH3P5+EUpcCvmc+mWQyGQbgj8IspO7LhPV23bHLbEMgHk5w7aUWqdSTqqqxH6yPia3GTAo28IMAebTvjqIYFQCPdQoVgZhvGBua5ZKpWqpTouYUwRKoV/2z0wxB2iWzFryby1+g8NBqN2zZL1bkSiAVaf0oekLe+ulQLEJ1t6dkoLDSEHd69NegcIOiUeAG/88QwHU778o0GQLMFq4LmiS8kvS9VD8jn2+nFq2ZqaL4fnP4picpJiADd4aQx4XteJuhWuUq9jtOlE+wgw+8aLst7gAAAABJRU5ErkJggg==)
Hence, the value of X is 5cm and Y is ![](data:image/png;base64,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)
Question 5.Prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side (Using basic proportionality theorem).
Answer:To prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side
⇒ Let us assume
ABC where DE is parallel to BC and D is the midpoint of AB.
![](data:image/png;base64,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)
Proof:
In
ABC, DE||BC
∴ AD = DB
Since, D is the midpoint of AB
⇒
……..eq(1)
⇒ now we know that basic proportionality theorem if a line drawn to one side of a triangle intersects the other two sides in distinct points, then it divides the other 2 side in the same ratio.
⇒ ![](data:image/png;base64,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)
⇒ 1 =
………eq(2)
From eq(1) and eq(2)
⇒ EC = AE
⇒ E is the midpoint of AC
Hence proved.
Question 6.Prove that a line joining the midpoints of any two sides of a triangle is parallel to the third side. (Using converse of basic proportionality theorem)
Answer:To prove that a line joining the midpoints of any two sides of a triangle is parallel to the third side.
⇒ now we know the converse of a basic proportionality theorem is if a line divides any two sides of a triangle in the same ratio then the line must be parallel to the third side.
⇒ Let us assume
ABC in which D and E are the mid points of AB and Ac respectively such that
⇒ AD = BD and AE = EC.
![](data:image/png;base64,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)
⇒ To prove that DE || BC
⇒ D is the midpoint of AB
∴ AD = DB
⇒
………eq(1)
Also, E is the midpoint of AC
∴ AE = EC
⇒
……..eq(2)
From equation (1) and (2) we get
⇒ ![](data:image/png;base64,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)
∴ DE || BC by converse of proportionality thereom
Hence, the line joining the mid points of any two sides of a triangle is parallel to three sides.
Question 7.In the given figure, DE||OQ and DF||OR. Show that EF||QR.
![](data:image/png;base64,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)
Answer:Given, DE ||OQ and DF || OR
Need to prove that EF || QR
⇒ Let us consider Δ POQ
⇒ By Basic proportionality theorem we have if a line drawn to one side of a triangle intersects the other two sides in distinct points, then it divides the other 2 side in the same ratio.
∴
………………………eq(1)
⇒ Consider Δ POR
⇒ DF || OR
∴
…………………eq(2)
⇒ From eq(1) and eq(2) we have
⇒ ![](data:image/png;base64,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)
∴ By converse of basic proportionality theorem we have if a line divides any two sides of a triangle in the same ratio then the line must be parallel to the third side.
∴ EF || QR
Hence proved
Question 8.In the adjacent figure, A, B, and C are points on OP, OQ and Or respectively such that AB||PQ and AC||PR. Show that BC||QR.
![](data:image/png;base64,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)
Answer:Given, AB || PQ and AC||PR
Need to prove BC || QR
⇒ In ΔOPQ, AB || PQ
⇒ Since, line drawn parallel to one side of triangle, intersects the other two sided in distinct point, then it divides the other 2 sides in same ratio.
⇒
……………eq(1)
⇒ In
OPR, AC || PR
⇒
……………eq(2)
From eq(1) and (2)
⇒ ![](data:image/png;base64,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)
Thus in
OQR, ![](data:image/png;base64,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)
⇒ Line BC divides the triangle OQR in the same ratio
⇒ We know that if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
∴ BC || QR
Hence proved.
Question 9.ABCD is a trapezium in which AB||DC and its diagonals intersect each other at point ‘O’. Show that ![](data:image/png;base64,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)
Answer:Given, ABCD is a trapezium where AB||DC
And diagonals intersect at each other ‘O’.
Need to prove ![](data:image/png;base64,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)
⇒ Let us draw a line EF||DC passing through point O.
![](data:image/png;base64,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)
⇒ Now, in
ADC, EO || DC
(because EF || DC)
So,
…………………eq (1)
Now, Line drawn parallel to one side of a triangle intersects the other two sides in distinct points, and then it divides the other 2 sides.
Similarly, in
DBA, EO || AB
(Because EF || AB)
⇒
…………………..eq(2)
⇒ From eq(1) and eq(2)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEMAAAAfCAMAAACLWIDNAAAAAXNSR0IArs4c6QAAAHVQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OgBmOjo6OjpmOjqQOma2OpC2OpDbZgAAZjoAZpDbZrb/kDoAkNv/tmYAtmY6tpA6tpBmttv/tv//25A627Zm29v/2////7Zm/7aQ/9uQ/9u2/9vb//+2///bcgDxogAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABPElEQVRIS+WV3VKDMBCFN1jqby3aNNU2lZZK3v8RPSeZEvSCrFMvHN0LBkLOye4y+yHyQxHsnE7h9drM1kXPvjHGVM9fBP6OHsHWW/FXu6KJr1vZY99Y0D3ssSoHyrubciJuKeEFp44Ewa74FOwSGSg8Yi3YOxa4uXS3O+mbFTyYZyG4+Yh0R4IDXE21Thn0DZOZDo8yqM+CU4MOc4nyky2nwRLCBsVngTV1G6zBWvdkqkWxEvHM+37L3ukEpbr+1XvH7sXAp5+KYd9wU+jTtwUX911dy8Un/TkDYtQ8qnGKeT0PXQawmwGjmGctTsmpOPxZEamIUONUwNNInEGRqPiZjtOfKzIQl8zTRMVkrMJphC9BmBVnkKtxmrDt6jYrSMXg9DiNv4T3DSZ5BGBQMeJRS0fytFq8/UKefgCivx2cQqqPngAAAABJRU5ErkJggg==)
⇒ ![](data:image/png;base64,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)
Hence Proved.
Question 10.Draw a line segment of length 7.2 cm and divide it in the ratio 5 : 3. Measure the two parts.
Answer:Need to draw a line segment with length 7.2cm
Also, to divide it 5:3 parts. Measure them.
⇒ Let m = 5 and n = 3
Construction steps:
1) Draw a ray AX, making an acute angle with AB.
2) Locate 8 = m + n points A1,A2,A3,A4,A5,A6,A7,A8 on AX such that AA1 = A1A2 = A2A3 = A3A4 = A4A5 = A5A6 = A6A7 = A7A8
3) Join BA8
4) Through the point A5 (m = 5) draw a line parallel to A8B at A5 intersecting AB at the point C. Then AC:CB = 5:3
![](data:image/png;base64,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)
⇒ Let the ratio be 5x:3x
⇒ 5x + 3X = 7.2cm
⇒ 8x = 7.2cm.
⇒ x = ![](data:image/png;base64,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)
⇒ x = 0.9cm.
Substituting ‘x’ value in 5x,3x we get
⇒ 5x = 5 × 0.9
= 4.5cm
⇒ 3x = 3 × 0.9
= 2.7cm.
In Δ PQR, ST is a line such that and also ∠PST = ∠PRQ. Prove that ΔPQR, is an isosceles triangle.
Answer:
Given,
Also, ∠PST = ∠PRQ
⇒ Need to prove that PQR is an isosceles triangle.
⇒ If a line divides any two sides of a triangles in the same ratio then the same line is parallel to the third side.
⇒ Since, ST || QR
⇒ ∠PST = ∠PQR …………eq(1)
(Corresponding angles)
Also given ∠PST = ∠PRQ ……………eq(2)
From eq(1) and eq(2) we get
⇒ ∠PQR = ∠PRQ
Since, sides opposite to equal angles are equal
⇒ PR = PQ
∴ two sides of PQR is equal
PQR is an isosceles triangle
Hence proved.
Question 2.
In the given figure, LM || CB and LN || CD Prove that
Answer:
Given, LM||CB and LN||CD
Need to prove
⇒ In ACB, LM || CB
⇒ now, we know that line drawn parallel to one side of the triangle, intersects the other two sides in distinct points, then it divides the other 2 side in same ratio.
∴ ……….eq(1)
And in ACD, LN||CD
⇒ …………eq(2)
From eq(1) and eq(2), we get
⇒
⇒
Adding 1 on both sides we get
⇒ + 1 =
+ 1
⇒ =
⇒ =
⇒ =
Hence, proved.
Question 3.
In the given figure, DE||AC and DF||AE Prove that
Answer:
Given, DE || AC and DF || AE
Need to prove
⇒ In ABC, DE || AC
⇒ now, we know line drawn parallel to one side of triangle, intersects the other two sides in distinct points, then it divides the other 2 side in same ratio.
⇒ ………eq(1)
⇒ In AEB, DF || AE
⇒ ………eq(2)
From (1) and (2)
⇒
Hence proved.
Question 4.
In the given figure, AB||CD||EF. given AB = 7.5 cm, DC = ycm, EF = 4.5cm, BC = x cm. Calculate the values of x and y.
Answer:
Given, Ab = 7.5cm, DC = ycm, EF = 4.5cm and BC = xcm
Need to calculate values of x and y
⇒ Let us consider Δ ACB and Δ CEF
⇒ Both are similar triangles
∴
⇒ x = 5cm
⇒ let us consider Δ BCD and Δ BFE
⇒ from basic proportionality theorem we have we know that line drawn parallel to one side of the triangle, intersects the other two sides in distinct points, then it divides the other 2 side in same ratio.
⇒
⇒ y =
⇒ Y =
⇒ Y =
Hence, the value of X is 5cm and Y is
Question 5.
Prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side (Using basic proportionality theorem).
Answer:
To prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side
⇒ Let us assume ABC where DE is parallel to BC and D is the midpoint of AB.
Proof:
In ABC, DE||BC
∴ AD = DB
Since, D is the midpoint of AB
⇒ ……..eq(1)
⇒ now we know that basic proportionality theorem if a line drawn to one side of a triangle intersects the other two sides in distinct points, then it divides the other 2 side in the same ratio.
⇒
⇒ 1 = ………eq(2)
From eq(1) and eq(2)
⇒ EC = AE
⇒ E is the midpoint of AC
Hence proved.
Question 6.
Prove that a line joining the midpoints of any two sides of a triangle is parallel to the third side. (Using converse of basic proportionality theorem)
Answer:
To prove that a line joining the midpoints of any two sides of a triangle is parallel to the third side.
⇒ now we know the converse of a basic proportionality theorem is if a line divides any two sides of a triangle in the same ratio then the line must be parallel to the third side.
⇒ Let us assume ABC in which D and E are the mid points of AB and Ac respectively such that
⇒ AD = BD and AE = EC.
⇒ To prove that DE || BC
⇒ D is the midpoint of AB
∴ AD = DB
⇒ ………eq(1)
Also, E is the midpoint of AC
∴ AE = EC
⇒ ……..eq(2)
From equation (1) and (2) we get
⇒
∴ DE || BC by converse of proportionality thereom
Hence, the line joining the mid points of any two sides of a triangle is parallel to three sides.
Question 7.
In the given figure, DE||OQ and DF||OR. Show that EF||QR.
Answer:
Given, DE ||OQ and DF || OR
Need to prove that EF || QR
⇒ Let us consider Δ POQ
⇒ By Basic proportionality theorem we have if a line drawn to one side of a triangle intersects the other two sides in distinct points, then it divides the other 2 side in the same ratio.
∴ ………………………eq(1)
⇒ Consider Δ POR
⇒ DF || OR
∴ …………………eq(2)
⇒ From eq(1) and eq(2) we have
⇒
∴ By converse of basic proportionality theorem we have if a line divides any two sides of a triangle in the same ratio then the line must be parallel to the third side.
∴ EF || QR
Hence proved
Question 8.
In the adjacent figure, A, B, and C are points on OP, OQ and Or respectively such that AB||PQ and AC||PR. Show that BC||QR.
Answer:
Given, AB || PQ and AC||PR
Need to prove BC || QR
⇒ In ΔOPQ, AB || PQ
⇒ Since, line drawn parallel to one side of triangle, intersects the other two sided in distinct point, then it divides the other 2 sides in same ratio.
⇒ ……………eq(1)
⇒ In OPR, AC || PR
⇒ ……………eq(2)
From eq(1) and (2)
⇒
Thus in OQR,
⇒ Line BC divides the triangle OQR in the same ratio
⇒ We know that if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
∴ BC || QR
Hence proved.
Question 9.
ABCD is a trapezium in which AB||DC and its diagonals intersect each other at point ‘O’. Show that
Answer:
Given, ABCD is a trapezium where AB||DC
And diagonals intersect at each other ‘O’.
Need to prove
⇒ Let us draw a line EF||DC passing through point O.
⇒ Now, in ADC, EO || DC
(because EF || DC)
So, …………………eq (1)
Now, Line drawn parallel to one side of a triangle intersects the other two sides in distinct points, and then it divides the other 2 sides.
Similarly, in DBA, EO || AB
(Because EF || AB)
⇒ …………………..eq(2)
⇒ From eq(1) and eq(2)
⇒
⇒
Hence Proved.
Question 10.
Draw a line segment of length 7.2 cm and divide it in the ratio 5 : 3. Measure the two parts.
Answer:
Need to draw a line segment with length 7.2cm
Also, to divide it 5:3 parts. Measure them.
⇒ Let m = 5 and n = 3
Construction steps:
1) Draw a ray AX, making an acute angle with AB.
2) Locate 8 = m + n points A1,A2,A3,A4,A5,A6,A7,A8 on AX such that AA1 = A1A2 = A2A3 = A3A4 = A4A5 = A5A6 = A6A7 = A7A8
3) Join BA8
4) Through the point A5 (m = 5) draw a line parallel to A8B at A5 intersecting AB at the point C. Then AC:CB = 5:3
⇒ Let the ratio be 5x:3x
⇒ 5x + 3X = 7.2cm
⇒ 8x = 7.2cm.
⇒ x =
⇒ x = 0.9cm.
Substituting ‘x’ value in 5x,3x we get
⇒ 5x = 5 × 0.9
= 4.5cm
⇒ 3x = 3 × 0.9
= 2.7cm.
Exercise 8.2
Question 1.In the given figure, ∠ADE = ∠B
i. Show that ΔABC ~ ΔADE
ii. If AD = 3.8cm, AE = 3.6 cm, BE = 2.1cm, BC = 4.2cm. find DE.
![](data:image/png;base64,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)
Answer:(i) Given, ∠ADE = ∠B = θ,∠C = 90°
⇒ ∠A = 90° -θ
⇒ if ∠A = 90°- θ, ∠B = θ
⇒ ∠ AED = 90°
⇒ now, comparing Δ ABC with Δ AED we have
∠ A common in both triangles
∠ C = ∠ AED = 90°
∠ ADE = ∠ B
∴ By AAA property we have in two triangles if the angles are equal, then sides opposite to the equal angles are in the same ratio (or proportional) and hence the triangles are similar.
Hence, Δ ABC and Δ ADE are similar triangles.
(ii) Given, Ad = 3.8cm, AE = 3.6cm, BE = 2.1cm, BC = 4.2cm
Need to find DE.
As Δ ABC and Δ ADE are similar triangles we have ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHMAAAAfCAMAAADjqwL1AAAAAXNSR0IArs4c6QAAAKhQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZjqQZpDbZrbbZrb/kDoAkDo6kGY6kGaQkLbbkNv/tmYAtmY6tpA6tpBmttv/tv//25A625Bm27Zm27aQ29uQ29u229vb29v/2////7Zm/7aQ/9uQ/9u2/9vb//+2///bsSbB8QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACIUlEQVRIS+2W61KDMBCFk9YLWrXYatUab3gpotUiSN7/zTy7gYijZeOMMx0d84NS2M232U3Yo9SKhjWbIFdjrXXvJCCGIjp1VvNDreMAB+UIyt5GWu+zQ7rDT9KNhcr6M3EOezZwzOrgUmW9K9HBE5L1O5WOyL7Yy0BTKhkpe8P07pFOTb1Oct4KYNaE3C/ImlP+w7nlILpHHsPDm7iwu0dNsMbbJpuq2AaTLi9y1NXRzJrpBSUGI4slICXQEapxE2qO1WkqSoq0vj9eOlVK9pqLAZ8QZEPwZSjH2KmEopXb64AthO1HuaV65EDm73n+Ok5PqMYgJDA3CNka3Z9x/Lt3AalSjxHx+jPeAVpgYvKaoAocrTBCSBT/Nr8gAwmfiuYoBQQc6uDnbW6kuT85aMHj2wQpgp9+H5oqz/22w09H/D/fajJwP/AiqMQ3eFg3xuXB8JedPtXuEy93ohYBDXcyVPbMaxrqZiX6jTSgm1zTIwFV7ElBtgholvu1ed1MuWGjp0sDusk1d9ykAWoI8zXtmnQBj/J8yr/WxIvS64elZMbRhXIrZ/YjYTjZGeJJotccU1E9YylVtXiCcILGsY0u6syNJxRR/DQ3bPvgNHJ1OG3Eb9cUVEWV4EI31bEYY5twQLoLuhgBgwmpUUQhTNppr9cQbSwdA6Rmi0DVs1jnfcRnhSRVBmUvnhXSTb34GThuGJICwypbBK6etEf/yvs3Fw03WgUxl3wAAAAASUVORK5CYII=)
⇒ AE + BE = 3.6 + 2.1 = 5.7
⇒ ![](data:image/png;base64,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)
⇒ DE = 4.2 × ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAeCAMAAADXVfSyAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6ADo6OgAAOgA6OjoAOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDo6kGY6kLbbkNv/tmYA25A625Bm27Zm27aQ29u2/7Zm/9uQ/9u2//+2///bQHBKxAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAYUlEQVQYV4WOSRaAIAxDU8VZcUQF8f7XtIC6YUE2v0PaF+DT2RFVgG1n7Nnip6YI3GqPndeuC9AMLWEbYsn/TapwblbKFu9VSVkP3GOU75oGZ18p9wQOwbnaBYYJJfzdqweIiQNraxZGfAAAAABJRU5ErkJggg==)
= 2.8cm
Hence, the value of DE is 2.8cm
Question 2.The perimeters of two similar triangles are 30 cm and 20 cm respectively. If one side of the first triangle is 12 cm, determine the corresponding side of the second triangle.
Answer:Given, perimeters of two similar triangles are 30cm and 20cm
And one side of the triangle is 12cm.
Need to find out the side of the second triangle.
⇒ since, the triangle are similar
⇒
= ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ x = ![](data:image/png;base64,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)
⇒ x = 8cm.
Hence, the corresponding side of the second triangle is 8cm
Question 3.A girl of height 90 cm is walking away from the base of a lamp post at a speed of 1.2m/sec. If the lamp post is 3.6 m above the ground, find the length of her shadow after 4 seconds.
Answer:Given, a girl with height 90cm and speed 1.2m/sec
And the lamp post is 3.6m above
Need to find the length of her shadow after 4sec
![](data:image/png;base64,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)
Consider Δ ABC and Δ DEC
⇒ ∠ ABC = ∠ DEC = 90°
⇒ ∠ ACB = ∠ DCE
⇒ Δ ABC ∼ Δ DEC
⇒ By converse proportionality theorem if a line divides any two sides of a triangle in the same ratio then the line must be parallel to the third side.
⇒ we have ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ x + 4.8 = 4x
⇒ 4x – x = 4.8
⇒ 3x = 4.8
⇒ X = ![](data:image/png;base64,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)
⇒ X = 1.6m
CE = 1.6m
Hence, the length of her shadow after 4 sec is 1.6m
Question 4.Given that Δ ABC ∼ Δ PQR, CM and RN are respectively the medians of Δ ABC and Δ PQR Prove that
i. Δ AMC ∼ Δ PNR
ii. ![](data:image/png;base64,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)
iii. ΔCMB ∼ ΔRNQ
![](data:image/png;base64,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)
Answer:(i) Given, Δ ABC ∼ Δ PQR
So,
………eq(1)
And ∠ A = ∠ P, ∠ B = ∠ Q and ∠ C = ∠ R ……..eq(2)
⇒ As CM and RN are medians
⇒ AB = 2AM and PQ = 2PN
From eq(1) we have
⇒ ![](data:image/png;base64,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)
i.e.,
……..eq(3)
Also, from eq(2) Δ MAC = Δ NPR ……eq(4)
⇒ From eq(3) and eq(4) we have
⇒ Δ AMC ∼ Δ PNR …..eq(5)
⇒ By SAS similarity if one angle of a triangle is equal to another angle of a triangle and the including sides of the these angles are proportional, then the two triangles are similar.
(ii) From eq(5) we have
……..eq(6)
⇒ From eq(1) we have,
⇒
…….eq(7)
⇒ From eq(6) and eq(7) we have
⇒
…..eq(8)
(iii) Again from eq(1) we have
⇒ ![](data:image/png;base64,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)
⇒ From eq(8) we have
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
i.e.,
…..eq(10)
⇒ From eq(9) and eq(10) we have
⇒ ![](data:image/png;base64,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)
∴ Δ CMB ∼ Δ RNQ
Question 5.Diagonals AC and BD of a trapezium ABCD with AB||DC intersect each other at the point ‘O’. Using the criterion of similarity for two triangles, show that ![](data:image/png;base64,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)
Answer:Given, trapezium ABCD and diagonals AC and BD intersect each other.
Need to prove ![](data:image/png;base64,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)
⇒ Let us consider Δ AOB and Δ DOC
⇒ ∠ AOB = ∠ DOC (vertically opposite angles)
⇒ by alternative interior angles we have
⇒ ∠ OAB = ∠ OCD
⇒ ∠ OBA = ∠ ODC
⇒ By AAA similarity we have in two triangles if the angles are equal, then sides opposite to the equal angles are in the same ratio (or proportional) and hence the triangles are similar.
⇒ Δ Aob ∼ Δ DOC
⇒ ![](data:image/png;base64,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)
Hence proved.
Question 6.AB, CD, PQ are perpendicular to BD. AB = x, CD = y and PQ = z prove that ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARwAAACOCAIAAABsTyBLAAAAAXNSR0IArs4c6QAAFspJREFUeF7tnWmUFdW1x80zfhDIWobuJYIy09AyCgIh0LSNNqKtgCAyk4BEmTIIrkTNyxNw5UUGGRMBbVBe2kYQHgpIMzTI0IAs0syDQIMgY0iayQgkyw++Hzm+Q1n39q2qe+sOVbXv6nXXvbdPnTr7f86uvc8+e/jeN998c5u8BAFBwD0E/sO9rqQnQUAQuImAA6ZaVVQkmAkCgoAlAnaZ6saNG5MnT9y1c6dlj9JAEAg4AnaZasWK5cePHpk//92A4yXkCwKWCNhiKsTUlpIS+lq2dMnp06ctO5UGgkCQEbDFVBs3bBg7bnyH7ByQWrtmTZDxEtoFAUsEbDHVuXPn0tLSBg36Cd0VFhZcvHjRsl9pIAgEFgFrpjpy+HDLli0BKKdTJ97ZWa1fvy6weAnhgoAlAt+zPPydNnVqZmbmP776xw+q/GD79u3z571Np4ePnbjzzjste5cGgkAAEbBgKswSJ0+c6JidraDBYpHZoC4f5uS/83heXgDxEpIFAUsEIql/sNDc/PzKlSvrXpBO3Xv24mtBwZ9lZ2UJrjQIJgKRJBVsc/36dXCp9O8XPHbt2jUjTDVr1gwmakK1IBABgUiSCosfbMOLDzNnzmjZrPGCBYWXLl1SPwpHycISBMIiYG39U5ddvXq1Tr0GRR9/3C2vS59nen300Ydwl2AqCAgCoQjYZSqu7NCx4+q1xQsXL01LT//VqBEIrokTJ+zdu1dgFQQEASMCDpiKyzBU/Lh9+1mz52zbUfry717VgmtdcTE7LkFWEBAEQMAZU2nI7r33vhEjRmrBNXTwIEzts2fPKisrE1gFgYAjYH34qwD67W9f4f0Pf3g9LF7w0rp1xRN+/xr/bdc+67nnh3XIypLT4YCvrcCSH6WkMuGVkZGB4MLNYt78Av4lgiuw60kIj179C4sdoim3c+dFi5es21jCjgvBlZvTEVOh7LhkqQUKAXcklU3BJabCQK2twBLrzp4qMnymHVfX7t3z8p6oWrVqYEEXwv2NQFwkVUWCS51x/edLv+aMa+SI4Z9u2yaGeH8vr2BSlwimUsjqM67d+w/998TJF8vL+z7TUxniRS0M5uLzK9WJUP8qwg5e2rZtqzbEi1ro10UWNLoSJ6lCkW3RooUyxKMW1s/I0GqhWAuDtgp9Rm8ymcqoFnKsjFo4483ZqIXqmEscC3221IJDTvKZSmONPfCpp3roYy7lWPhQx6z33isQj/jgrEgfUJrMPVVk+DAM7tm9e8XHKwr/52YGzwE/HdL1ya648/oAdCHB3wikkKQyAa2shUotxFq4taQEa6EILn8vR39Ql7pMZVQLBw4ctKlkC/YMYrqUPQMHX465/DEHQoXPEPAAU2nERXD5bPH5lRwvMZWaA+wZIrj8uhz9QZf3mEoElz9Wno+p8DBTieDy8br0NGmeZyoRXJ5ef74cvH+YSgSXLxeoF4nyG1OJ4PLiKvTZmH3LVCK4fLZSPUSOz5lKBJeH1qJvhhoUpooguCQG2TerOUUICRZThRVcxhhkSQaaIuvS08NIXS/1RMJqikHuN2BAdvZDkpomkVPgp3sFVFKZptAYg6zLLyifXUlN46flnhhaRFKFwfns2TMbNmzIf+utk58fo4BQ3/79c3M7k4U3MVMid/E6AiKpwswg5ReUz+7yojV5Tz6pU+2K4PL6ck/M+EVSWeOMBrh1y5b8t9/avm2LElzdunWD8ayvlBb+QoC68ksWL16xYlnTps3q1q2XmZlZrVq1+xs3NtXiEEllPe06R7wWXO3btpYoSWvg/NXigw8WZbVrA00ri1bP/OOfRo8ZU69evTfemKzqYn/n9Y291yuvvMyfvbY+b0V98Q8/XJqd1aFWjWq8FxT8mV98TnPgyStauZLp7te3jwmJzZs2wVSmH0VSOX6cqqxPpvB+SajmGEdPXcCjk/E+1aOHadSt29yUXaaXMFX0c6vD+42VWqXEePSApuqVbKW2bt7I6BrUb2AaI1uD0OKGwlSxziSCy1ipVUqMxwpo6l1/8sQJR4MSpnIEV4WNw5YYl4Rq7oCb7F7q1K2rhnDhwgU7YxGmsoOSgzbGEuOSUM0BcCnctGbNmh2ycxjguXPn7AxTmMoOSo7bSCZQx5Cl9gWDBv2EARYWFpjc1nbt3Bk68NvHjRtnh5z169fT7JFHcu00ljYaAbirefMWQ559tt2P29/41z+nvzHprdlvlh07xk4sPT39jjvuEKw8gQBOapn3N6by0769ezmegrW+/PLLNWtWf/3116H+a+JRkdA5pdJCUdHKFcuWiXNGQnF36Wbw0meHDrGzqlKlCl1iTw81/fG7MJVLeDvshmCT1atXzZo5neue6PYU2sUDLVuGnSGHHUtzNxGAi86cOXPw4IEdO3YMGfKsTadqYSo358BpXwiuzZs3vV9YqAWXuMM7xdD19jDS0aNH9+/ft23r1pXLP9L9UzyNQ387tzMbKuiRwAd9pRSGsgNi1G10SS5xh48aQ1cuZJ0TgkD56T7P9KLgIIXRqINBz5SbYWoo9unoLt+RVJRXU31RvebGjev9+/YloOiLs3/lF/xHeaewjaPepbEjBEzu8KNffLFLl8dEJ3SEof3GCI/PDn22c9dOyguyztWFlEFr27ZtkyZNTZpe7XvvsS+pzOofnIoqAoOePXv26tWrmpGEqezPVuwt2XEtWrRQVbvDB6pPn74S2x87qvRADhK1QVLY8iKWh5C5B1s9eH/j+yOE88TEVEpYcacJEycZyxYKU7kyqY46YQWsW1dMiCRXjfzlCz17Pm1zo+zoLv5uHHaD1K59Vs7DDz/Q4oFGmZk2n1YxMRUTmZvTEaCV1qdfwlTJWnyo+0ithQsWoKJIjVY7s4Bed+qLU3v27tm/b5+2NGBibd+hQ7NmzXGPsMlIxnvFxFR0xPW8r9tYYnwuClPZmc74teGJy1HjtClTYC0etC+MHiMmeI220fBNGdvQDdJ9990X49bUEVN9a/3jcbiuuJjBYQBhzhjuu+++w1cCGeK3UKRn+wiwJnQQF1dR/vixRzszWUZTrf3eKmpJjMO0qVNZQPzxoX+/vr/8xc9LNm+OvWfXe4BwVqy216FeER9wvKyMZAfz5hds21GKqoVdDdCQDTFylOPBq6BFQlkJbOQP2/yZM6fV5xHDh/FZNZDI35QKft2zZ8+ECa/raWLWQuNPoxtweXm56vbUqVP0QKwrn+fm50fXm4tXEV4N1SxUlqIaoYq85is/8i+3EAg7Zu7FXWyS8631j63U0qX/O3DgQGUAQUBhIRk1apS2h4j65/hxFf8LQs+OY89Ig7BSmRi2bP8L2495c+e+NvZ3oXvs+BN38w4syy9OnsTqvau0FKO0uim7o2bNm2NmqFW7VsLS7zhS/8SjIjHLI753cdHpCYHQqnkTzVQogdOnTKrfsNEnGzbFl4Z/985j4sjhw8c/P37o0CGT1btRo0YcH8W+O4qOCmGq6HDz/FWuCC6jpKpUqdIzvXoeP3pk7fqNWJ/jBJA6Ozpy5IjxEFYZ6+rXq2/f6h2n4aluhaniCq8HOjcKLqdWeC2pXh3/e0itUaNG2x/9KC0tzUWyw5q8MY+1at2aQ9jadeqk4HGcMJWLC8DDXSnBpa3wVF2w4/Rk2lO5Qr86gT1x4nM26qEmb7JSRnd25MrYbHYiTGUTqKA0w1WUDFvqGNTS6cm0p4oaI+VZV3asbOMnn2gbg3JlyGiQEdknKOqbxu9CYar4YevhnpWBV0VwoRPiT0itk1B6opZURnFktDGQqEO5qCbLxuDKnAlTuQKjPztRoceqoAly47nnh3XIytJno4ipQwcPDuzXG+Ln5L+T06lT5GPTsOIoKSbveM+WMFW8EfZ8/0iVPbt3T582VQdHKkd4fucIFaOfeodOE1P5WxxFmFdhKs8v+oQRYEcnrMg/NfEnsAmDJfRGwlRJBN+TtzbphNgS8O9+tEuXiox1DRs2TLQ3XbJxFaZK9gx47f4wFfYJsjKouG/9UsW4vGisc30GHDGVJNN0HX8PdMjWCMUPD0+KlZCbumWzxiorA6YLoiGxDSoaMGZgEK9cuXLVqm4e/noAoNiGKEwVG37eufpmHHFxMVykcpuoWAkcVQkmJ/sC4XOkN1m0eMlLL72MBRyy+EoMBR90mIlkAbI52+JQaxMo7zWryFIHF0V2TkWCwW869DsWpyfvoVbBiB2pf8JUvpn3m4QoL+/QSHJlqbPpnGpiKgWQyZiB01N29kNRxKV7FG5hKo9OXJTD1kFHoam2cKuLwlIXlqn04IxOT8FJR+OIqWRPFeVSTuJl6HWoZCx9IkeZbHZHQwcPgqPQ69gFsTvSkeQ4Irlu+ybH1qzZc4hXV/UjuTubNJWLIYmYJPLWq4qKyDJAXW1uyjtJB0y0C1MlcjqivxcnsCqF6sgRw1UKVbY9Vy5fZmUvXLyU5KfUIMbGkNu5s7txE0T+qpQV/I199VVNgC7DpYwZcDWjcj1nRvR4xfNKYmGWLV1y8MBBblL9nuqqcqnxJXuqeMIfQ986Q5Apek+nUI2ff6pR/eORjA4JHQRuTJw0uSK5Z0xRqOotGJNGxgBDqlxqUv8e7vTQPfdUX/D+QsZHcPToMWOEqVJlqkzjULuj0HAJFb2XsHAJzVRKq4GRSKjUuEkTy1BFU+jxc8OG5eU94Q9jhompeNYgrFCzOTS/fu2aKSxa1L9kMpg6O1JKnd4dkTezfkYGZ0ekxkevU2dH6HUJS3KiEVGl1zEn3n333ZYcxVW63gIaqbE0KzvAZKIcz3sf2L8/NNGAMFU8IQ/Xt3JlMNoYSOx81w9/qApMqN2RSliHmSHpj3mCQa6FPIktIUP3gwRogShCfdkBYsyAap8ZM7BYEB0TioYwleUKcaGBiZGUjYEFxxNdp30cOHBQKnCRkVp4gMQvx47frIiBBgiDOcKCJwJErV5bDJlp6elQ7TNjBnVKw28ybeYHlGSaNoHSzcjKYMr8SHJSwtrjnfbR6ThD26vMqvxetHKlSqbJZ95J6R5L5wAya9abxgSgsfSW4GtNyTR/8fNREdAQ65+jh2+kxtrX21iBT+fFj+IQ1rWROezIaP1DOs2cMaNJ0yZbSkrGjhtvZ2cV+W4mYwY1uDzhmaEMFWlV087/9XzXrt0WFBb2HzCgIluoMJXDFffd5hWlUFUFJjzESEayIntUxISX4WKO3VZ8vELX4ErxuqyKqXbv2j1/3tskb4vAUZAoTOVskVh6qbp79upscC61TgxTqcFyqL18+XJVgyuVq5kopiLHG+O0dFIRQ4X1SlRuQZTDi+zNoApMWHcnLQwIKM8MU5gJUKdmmIk6Y7CcQJFU4SEKW4HPixskyxUQ2iCRksp0d2NdVrx1H3vs8bB51KIgKsZLHDnUClPdQjvIjJT4PVWEVW4sHhmaRy1G9ojuckdMFXT1j/kzOaqqPA3qKBa1BI9sdYJkR+5HN2FylQkBDrjQCdUBF//CW1dVuEtNnTB0+oIoqVQKSFPVowQ4qnqFeZKo/oWFCBMrdT2VnTBZEVyOJFVQmErVazHm3CJVEAFIKVtmIokcmGpMpaAw6YQJrnrsiKl8q/5pk532stM1YZVzUJwCkJLIDP6+tdYJjeloUtNO6CtJpevwhfVpSP16LSnCFakpqSLbCXv2fDqu5xmOJJXnmSpsjny1QYouQ0OKrOwkDsMTTJVgndARU3lS/YORVAo7EkG2b9sa6xAxSETyqfx1cc3QkMS1LrcORSA1dULXJBULfcOGDVif4zT3oTmDtKUhYSGxcSItwd2y25w//93Bg4dUdEjgIUkVQScke4dNf0I2ZlWqVIk8C2zIeWTjNGNnsmwxFXuVHt27kQTYskedgdGypZ0G2mSni4ipUnyksKtVu1biI2HtjDn123AuR9JZy3FyUhe/R6Tl3WNpYPQntJMwA9XOzu1cZqqb5wMjhl8sLyeFYkW3x1rN0o+dqSIwks1ckHYACnIbECavGA/y6tWrh8VBTSUGUk8/thDIa9asfr+wUNXgipAww85+yU6bW2DaDPayDFLUkW02OzQ2O3PmdPHatdxCRbDxRzwfAW0Y8YjniaJDuSQCAgQLgjDvFbWJZSpTEHlWkV5aEya8Hkq4KQAxLAl22ugLk2aoCDU2HC8r01nscA7CUYVUB0lP0hBkmeYP2nXCDGP2T3Tg+CXMSChTaUc70oAoq53Kq6rSQZI2SBjJH+s4BalIpJ3wW6YifRn5R1UiUpIDksmWPz7Ezs3KswFvSBiJOkhskZX5m3Nx7dYgEikFV6Evh4TBk2RvPL7xltZJ1KD0/PnzbtKrFcFTp06p/Qwf+HFufj6fO+VkqwaO9lRshEhvQpITtkZ6m0QPKOsRVPkUVMd9OaSg7akiTCILlVWqlmjvXk+zsaeCeBz3VJR8hXePHz2CELPPxFoiqcp8pLMzHcjG1ZfE/jilpSAAAiqJGh9U8UjLGBPyimqFjizz5Knlj+x/JjBv7akqVapk/N/JEyfU1/T09IomwLhH4nSMZiQbUNlVjZ4NwkiyglMcAZza0AlZtOzwWcPsU/DDxphhGjaHOt27d1c/Dv3Zz2b+8U+coU+ePJHU6sakiLeYCsGnWpPJlgIhY8f+1wsv/mbXvoOmc3cjI6k9kmIkxevaRUgYKcWXkQwvFAEWLRm2dWJd1rZlYt1WDz7YtWt3FLrx48bqDm95VMBqrZo34R9z8t/Ba2Px4g8OHNg/YMAgOJIfYVzy9/LB6FfBmRq7PVUiVp0YIqNktlIcATblPActD3+DM5UoWcoD2zRx6uBY/6g8Knbt3Nmj2xNKfqh/kbt3YL/efEACqbyIt5iKvVNWuzb8tGX7X4iSwGpHkl6+vvf+B02aNrXpppTi60mGJwhEjQD2aiyHoUylf1GMc7N/besoLy83Wv/4ndy2/ELuXztGMLRHLH52WkqbpCOAk0HSx+DRAewsLVVsUpHZnN/D7KkUKyOpqMDDh2rVqtlhbrZeKZJNys5oA96moopsbAFQZthRY9Hiz2lFgmCi+ve//U0Rrk193zIV8GGfUP/79NNtIDv02SH1GzaaPG0GW7FgghU0qrERs6kmV3jv3n0ez8vjYcpXfozdAcDfSF64cAECO2Tn3LKfKymG8ob6Z3r3qICWYUeBAEo+Kg0Kv/HaqVOmxF7sI4rBpPIloeqfKozC72aHWpQ3DBemd38/YIQ6IwIctvA1q2NH44+ZmZl8/fXoX4mwUrCg0KlqXbxQj1HoOPzNyGi4dv1Go0JnK0hR1p+/EdCGXyy9HbOzNbE4EDz6SA5fP1y+UnYB4MDDBW0ONU+/h60tlFAvdX8vTe9Sp71nKleubKSi0v9/VdsGeYUqdGExEaaSpXIbdbIVCpT3DQuHTQuwQKkQEKaSlXBbrdq1sfQCxFdffWWEQ0uw2nXqCEz2ERCmso+Vb1ui1QwfMRLyVq36jsN1aWkpP1I4MPaqpL7FLhxhwlSBmu4KieVsavDQ5znux+lGNcJ6MX3KpO49e1GKUzByhIBY/xzB5fPGcNSyZcsg8q677rpy5Upubm7rNm2khpDTWRdJ5RQxP7fHbj7+tdcuX76EjIJOzOvCUVHM9+3jxo2L4jK5xMcING3W7OKly4sWFFy5+uX3b7+dfI/Va9TwMb2ukybqn+uQ+qRDvAdU3CpnnWKocDSpwlSO4JLGgoA1ArKnssZIWggCjhAQpnIElzQWBKwREKayxkhaCAKOEBCmcgSXNBYErBEQprLGSFoIAo4QEKZyBJc0FgSsERCmssZIWggCjhD4PwFpfU44MO5hAAAAAElFTkSuQmCC)
Answer:Given, in Δ BCD, PQ || CD
….eq(1)
And in Δ ABD, PQ||AB
⇒
….eq(2)
Need to prove that ![](data:image/png;base64,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)
⇒ From eq(1) and eq(2) we have
⇒ ![](data:image/png;base64,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)
⇒ 1-
[FROM EQ(1)]
⇒ 1 = PQ(
)
⇒
…..eq(3)
Since, from the question we know that AB = X CD = Y and PQ = z
Substituting those values in eq(3) we get
⇒ ![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
Hence proved
Question 7.A flag pole 4 m tall casts a 6 m., shadow. At the same time, a nearby building casts a shadow of 24m. How tall is the building?
Answer:Given, a flag pole 4m tall with shadow 6m
And a building shadow 24m
Need to calculate the building
In Δ ABC and Δ PQR
![](data:image/png;base64,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)
⇒ ∠ B = ∠ Q = 90°
⇒ ∠ C = ∠ R
At any instance all sun rays are parallel, AC || PR
⇒ by AAA similarity we have in two triangles if the angles are equal, then sides opposite to the equal angles are in the same ratio (or proportional) and hence the triangles are similar.
⇒ Δ ABC ∼ Δ PQR
⇒ By Converse proportionality theorem we have
if a line divides any two sides of a triangle in the same ratio then the line must be parallel to the third side.
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ PQ = ![](data:image/png;base64,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)
= 16m
Hence the height of a building is 16m
Question 8.CD and GH are respectively the bisectors of ∠ACE and ∠EGF such that D and H lie on sides AB and FE of Δ ABC and Δ FEG respectively. If Δ ABC ∼ Δ FEG then show that
i. ![](data:image/png;base64,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)
ii. Δ DCB ∼ ΔHGE
iii. Δ DCA ∼ ΔHGF
Answer:![](data:image/png;base64,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)
Given, Δ ABC ∼ Δ FEG …..eq(1)
⇒ corresponding angles of similar triangles
⇒ ∠ BAC = ∠ EFG ….eq(2)
And ∠ ABC = ∠ FEG …….eq(3)
⇒ ∠ ACB = ∠ FGE
⇒ ![](data:image/png;base64,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)
⇒ ∠ ACD = ∠ FGH and ∠ BCD = ∠ EGH ……eq(4)
Consider Δ ACD and Δ FGH
⇒ From eq(2) we have
⇒ ∠ DAC = ∠ HFG
⇒ From eq(4) we have
⇒ ∠ ACD = ∠ EGH
Also, ∠ ADC = ∠ FGH
⇒ If the 2 angle of triangle are equal to the 2 angle of another triangle, then by angle sum property of triangle 3rd angle will also be equal.
⇒ by AAA similarity we have in two triangles if the angles are equal, then sides opposite to the equal angles are in the same ratio (or proportional) and hence the triangles are similar.
∴ Δ ADC ∼ Δ FHG
⇒ By Converse proportionality theorem
⇒ ![](data:image/png;base64,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)
Consider Δ DCB and Δ HGE
From eq(3) we have
⇒ ∠ DBC = ∠ HEG
⇒ From eq(4) we have
⇒ ∠ BCD = ∠ FGH
Also, ∠ BDC = ∠ EHG
∴ Δ DCB ∼ ΔHGE
Hence proved.
Question 9.AX and DY are altitudes of two similar Δ ABC and ΔDEF. Prove that AX : DY = AB : DE.
Answer:![](data:image/png;base64,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)
Given, Δ ABC ∼ Δ DEF
⇒ ∠ ABC = ∠ DEF
⇒ consider Δ ABX and Δ DEY
⇒ ∠ ABX = ∠ DEY
⇒ ∠ AXB = ∠ DYE = 90°
⇒ ∠ BAX = ∠ EDY
⇒ By AAA property we have in two triangles if the angles are equal, then sides opposite to the equal angles are in the same ratio (or proportional) and hence the triangles
⇒ Δ ABX ∼ Δ DEY
⇒ By Converse proportionality theorem we have
if a line divides any two sides of a triangle in the same ratio then the line must be parallel to the third side.
⇒ ![](data:image/png;base64,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)
∴ AX:DY = AB:DE
Hence proved
Question 10.Construct a triangle shadow similar to the given ΔABC, with its sides equal to
of the corresponding sides of the triangle ABC.
Answer:Given, a Δ ABC, we are required to construct a triangle whose side are of the corresponding sides of Δ ABC
Construction Steps:
1) Draw any ray BX making an acute angel with BC on the sides opposite to the vertex A.
2) Locate the points B1, B2, B3, B4, B5 on BX such that BB1 = B1B2 = B2B3 = B3B4 = B4B5.
![](data:image/png;base64,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)
3) Join B3 to C as 3 being smaller and through B5 draw a line parallel to B3C, intersecting the extended line segment BC at C’.
4) Draw a line through C parallel to CA intersecting the extended line segment BA at A’.
Then A’BC’ is the required triangle.
Justification:
Note Δ ABC ∼ Δ A’BC’
∴ ![](data:image/png;base64,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)
So, ![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
Question 11.Construct a triangle of sides 4 cm, 5 cm and 6 cm. Then, construct a triangle similar to it, whose sides are
of the corresponding sides of the first triangle.
Answer:![](data:image/png;base64,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)
Given, sides AB = 4cm, BC = 5cm, CA = 6cm.
Need to construct a triangle whose sides are
of the corresponding sides of Δ ABC.
Construction steps:
1) Making an angle of 30° draw any ray BX with the base BC of Δ ABC on the opposite side of the vertex A
2) Locate three points B1,B2,B3 on BX so that BB1 = BB2 = BB3, the number of points should be greater of m and n in the scale factor ![](data:image/png;base64,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)
3) Join B2 to C’ and draw a line through B3 parallel to B2C intersecting the line segment BC at C.
4) Draw a line through C parallel to C’A intersecting the line segment BA at A’. Then A’B’C is the required triangle.
Hence proved
Question 12.Construct an Isosceles triangle whose base is 8 cm and altitude is 4 cm. Then, draw another triangle whose sides are
times the corresponding sides of the isosceles triangle.
Answer:Given, base of a triangle as 8cm and altitude as 4cm
Need to draw an isosceles triangle
Construction steps:
![](data:image/png;base64,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)
1) Draw a line segment BC = 8cm
2) Draw a perpendicular bisector AD of BC
3) Join AB and AC we get an isosceles triangle Δ ABC
4) Construct an acute angle ∠ CBX downwards.
5) On BX make three equal arts.
6) Join C to B2 and draw a line through B3 parallel to B2C intersecting the line extended line segment BC at C’
7) Again draw a parallel line C’A’ to AC cutting BP at A’
8) Δ A’BC’ is the required triangle.
In the given figure, ∠ADE = ∠B
i. Show that ΔABC ~ ΔADE
ii. If AD = 3.8cm, AE = 3.6 cm, BE = 2.1cm, BC = 4.2cm. find DE.
Answer:
(i) Given, ∠ADE = ∠B = θ,∠C = 90°
⇒ ∠A = 90° -θ
⇒ if ∠A = 90°- θ, ∠B = θ
⇒ ∠ AED = 90°
⇒ now, comparing Δ ABC with Δ AED we have
∠ A common in both triangles
∠ C = ∠ AED = 90°
∠ ADE = ∠ B
∴ By AAA property we have in two triangles if the angles are equal, then sides opposite to the equal angles are in the same ratio (or proportional) and hence the triangles are similar.
Hence, Δ ABC and Δ ADE are similar triangles.
(ii) Given, Ad = 3.8cm, AE = 3.6cm, BE = 2.1cm, BC = 4.2cm
Need to find DE.
As Δ ABC and Δ ADE are similar triangles we have
⇒
⇒ AE + BE = 3.6 + 2.1 = 5.7
⇒
⇒ DE = 4.2 ×
= 2.8cm
Hence, the value of DE is 2.8cm
Question 2.
The perimeters of two similar triangles are 30 cm and 20 cm respectively. If one side of the first triangle is 12 cm, determine the corresponding side of the second triangle.
Answer:
Given, perimeters of two similar triangles are 30cm and 20cm
And one side of the triangle is 12cm.
Need to find out the side of the second triangle.
⇒ since, the triangle are similar
⇒ =
⇒
⇒ x =
⇒ x = 8cm.
Hence, the corresponding side of the second triangle is 8cm
Question 3.
A girl of height 90 cm is walking away from the base of a lamp post at a speed of 1.2m/sec. If the lamp post is 3.6 m above the ground, find the length of her shadow after 4 seconds.
Answer:
Given, a girl with height 90cm and speed 1.2m/sec
And the lamp post is 3.6m above
Need to find the length of her shadow after 4sec
Consider Δ ABC and Δ DEC
⇒ ∠ ABC = ∠ DEC = 90°
⇒ ∠ ACB = ∠ DCE
⇒ Δ ABC ∼ Δ DEC
⇒ By converse proportionality theorem if a line divides any two sides of a triangle in the same ratio then the line must be parallel to the third side.
⇒ we have
⇒
⇒
⇒ x + 4.8 = 4x
⇒ 4x – x = 4.8
⇒ 3x = 4.8
⇒ X =
⇒ X = 1.6m
CE = 1.6m
Hence, the length of her shadow after 4 sec is 1.6m
Question 4.
Given that Δ ABC ∼ Δ PQR, CM and RN are respectively the medians of Δ ABC and Δ PQR Prove that
i. Δ AMC ∼ Δ PNR
ii.
iii. ΔCMB ∼ ΔRNQ
Answer:
(i) Given, Δ ABC ∼ Δ PQR
So, ………eq(1)
And ∠ A = ∠ P, ∠ B = ∠ Q and ∠ C = ∠ R ……..eq(2)
⇒ As CM and RN are medians
⇒ AB = 2AM and PQ = 2PN
From eq(1) we have
⇒
i.e., ……..eq(3)
Also, from eq(2) Δ MAC = Δ NPR ……eq(4)
⇒ From eq(3) and eq(4) we have
⇒ Δ AMC ∼ Δ PNR …..eq(5)
⇒ By SAS similarity if one angle of a triangle is equal to another angle of a triangle and the including sides of the these angles are proportional, then the two triangles are similar.
(ii) From eq(5) we have ……..eq(6)
⇒ From eq(1) we have,
⇒ …….eq(7)
⇒ From eq(6) and eq(7) we have
⇒ …..eq(8)
(iii) Again from eq(1) we have
⇒
⇒ From eq(8) we have
⇒
⇒
i.e., …..eq(10)
⇒ From eq(9) and eq(10) we have
⇒
∴ Δ CMB ∼ Δ RNQ
Question 5.
Diagonals AC and BD of a trapezium ABCD with AB||DC intersect each other at the point ‘O’. Using the criterion of similarity for two triangles, show that
Answer:
Given, trapezium ABCD and diagonals AC and BD intersect each other.
Need to prove
⇒ Let us consider Δ AOB and Δ DOC
⇒ ∠ AOB = ∠ DOC (vertically opposite angles)
⇒ by alternative interior angles we have
⇒ ∠ OAB = ∠ OCD
⇒ ∠ OBA = ∠ ODC
⇒ By AAA similarity we have in two triangles if the angles are equal, then sides opposite to the equal angles are in the same ratio (or proportional) and hence the triangles are similar.
⇒ Δ Aob ∼ Δ DOC
⇒
Hence proved.
Question 6.
AB, CD, PQ are perpendicular to BD. AB = x, CD = y and PQ = z prove that
Answer:
Given, in Δ BCD, PQ || CD
….eq(1)
And in Δ ABD, PQ||AB
⇒ ….eq(2)
Need to prove that
⇒ From eq(1) and eq(2) we have
⇒
⇒ 1- [FROM EQ(1)]
⇒ 1 = PQ()
⇒ …..eq(3)
Since, from the question we know that AB = X CD = Y and PQ = z
Substituting those values in eq(3) we get
⇒
∴
Hence proved
Question 7.
A flag pole 4 m tall casts a 6 m., shadow. At the same time, a nearby building casts a shadow of 24m. How tall is the building?
Answer:
Given, a flag pole 4m tall with shadow 6m
And a building shadow 24m
Need to calculate the building
In Δ ABC and Δ PQR
⇒ ∠ B = ∠ Q = 90°
⇒ ∠ C = ∠ R
At any instance all sun rays are parallel, AC || PR
⇒ by AAA similarity we have in two triangles if the angles are equal, then sides opposite to the equal angles are in the same ratio (or proportional) and hence the triangles are similar.
⇒ Δ ABC ∼ Δ PQR
⇒ By Converse proportionality theorem we have
if a line divides any two sides of a triangle in the same ratio then the line must be parallel to the third side.
⇒
⇒
⇒ PQ =
= 16m
Hence the height of a building is 16m
Question 8.
CD and GH are respectively the bisectors of ∠ACE and ∠EGF such that D and H lie on sides AB and FE of Δ ABC and Δ FEG respectively. If Δ ABC ∼ Δ FEG then show that
i.
ii. Δ DCB ∼ ΔHGE
iii. Δ DCA ∼ ΔHGF
Answer:
Given, Δ ABC ∼ Δ FEG …..eq(1)
⇒ corresponding angles of similar triangles
⇒ ∠ BAC = ∠ EFG ….eq(2)
And ∠ ABC = ∠ FEG …….eq(3)
⇒ ∠ ACB = ∠ FGE
⇒
⇒ ∠ ACD = ∠ FGH and ∠ BCD = ∠ EGH ……eq(4)
Consider Δ ACD and Δ FGH
⇒ From eq(2) we have
⇒ ∠ DAC = ∠ HFG
⇒ From eq(4) we have
⇒ ∠ ACD = ∠ EGH
Also, ∠ ADC = ∠ FGH
⇒ If the 2 angle of triangle are equal to the 2 angle of another triangle, then by angle sum property of triangle 3rd angle will also be equal.
⇒ by AAA similarity we have in two triangles if the angles are equal, then sides opposite to the equal angles are in the same ratio (or proportional) and hence the triangles are similar.
∴ Δ ADC ∼ Δ FHG
⇒ By Converse proportionality theorem
⇒
Consider Δ DCB and Δ HGE
From eq(3) we have
⇒ ∠ DBC = ∠ HEG
⇒ From eq(4) we have
⇒ ∠ BCD = ∠ FGH
Also, ∠ BDC = ∠ EHG
∴ Δ DCB ∼ ΔHGE
Hence proved.
Question 9.
AX and DY are altitudes of two similar Δ ABC and ΔDEF. Prove that AX : DY = AB : DE.
Answer:
Given, Δ ABC ∼ Δ DEF
⇒ ∠ ABC = ∠ DEF
⇒ consider Δ ABX and Δ DEY
⇒ ∠ ABX = ∠ DEY
⇒ ∠ AXB = ∠ DYE = 90°
⇒ ∠ BAX = ∠ EDY
⇒ By AAA property we have in two triangles if the angles are equal, then sides opposite to the equal angles are in the same ratio (or proportional) and hence the triangles
⇒ Δ ABX ∼ Δ DEY
⇒ By Converse proportionality theorem we have
if a line divides any two sides of a triangle in the same ratio then the line must be parallel to the third side.
⇒
∴ AX:DY = AB:DE
Hence proved
Question 10.
Construct a triangle shadow similar to the given ΔABC, with its sides equal to of the corresponding sides of the triangle ABC.
Answer:
Given, a Δ ABC, we are required to construct a triangle whose side are of the corresponding sides of Δ ABC
Construction Steps:
1) Draw any ray BX making an acute angel with BC on the sides opposite to the vertex A.
2) Locate the points B1, B2, B3, B4, B5 on BX such that BB1 = B1B2 = B2B3 = B3B4 = B4B5.
3) Join B3 to C as 3 being smaller and through B5 draw a line parallel to B3C, intersecting the extended line segment BC at C’.
4) Draw a line through C parallel to CA intersecting the extended line segment BA at A’.
Then A’BC’ is the required triangle.
Justification:
Note Δ ABC ∼ Δ A’BC’
∴
So,
∴
Question 11.
Construct a triangle of sides 4 cm, 5 cm and 6 cm. Then, construct a triangle similar to it, whose sides are of the corresponding sides of the first triangle.
Answer:
Given, sides AB = 4cm, BC = 5cm, CA = 6cm.
Need to construct a triangle whose sides are of the corresponding sides of Δ ABC.
Construction steps:
1) Making an angle of 30° draw any ray BX with the base BC of Δ ABC on the opposite side of the vertex A
2) Locate three points B1,B2,B3 on BX so that BB1 = BB2 = BB3, the number of points should be greater of m and n in the scale factor
3) Join B2 to C’ and draw a line through B3 parallel to B2C intersecting the line segment BC at C.
4) Draw a line through C parallel to C’A intersecting the line segment BA at A’. Then A’B’C is the required triangle.
Hence proved
Question 12.
Construct an Isosceles triangle whose base is 8 cm and altitude is 4 cm. Then, draw another triangle whose sides are times the corresponding sides of the isosceles triangle.
Answer:
Given, base of a triangle as 8cm and altitude as 4cm
Need to draw an isosceles triangle
Construction steps:
1) Draw a line segment BC = 8cm
2) Draw a perpendicular bisector AD of BC
3) Join AB and AC we get an isosceles triangle Δ ABC
4) Construct an acute angle ∠ CBX downwards.
5) On BX make three equal arts.
6) Join C to B2 and draw a line through B3 parallel to B2C intersecting the line extended line segment BC at C’
7) Again draw a parallel line C’A’ to AC cutting BP at A’
8) Δ A’BC’ is the required triangle.
Exercise 8.3
Question 1.Equilateral triangles are drawn on the three sides of a right angled triangle. Show that the area of the triangle on the hypotenuse is equal to the sum of the areas of triangles on the other two sides.
Answer:Given, right angled triangle ABC with AC as hypotenuse
![](data:image/png;base64,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)
⇒ Let AB = a, BC = b, AC = c
⇒ we have a2 + b2 = c2 …………(1)
⇒ We know that area of equilateral triangle = ![](data:image/png;base64,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)
⇒ Area of ACD = ![](data:image/png;base64,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)
⇒ Area of BCF = ![](data:image/png;base64,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)
⇒ Area of AEB = ![](data:image/png;base64,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)
⇒ Area of AEB + Area of BCF = ![](data:image/png;base64,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)
⇒ From eq(1) we have a2 + b2 = c2
⇒ Area of AEB + Area of BCF =
= Area of ACD
Hence, the area of the triangle on the hypotenuse is equal to the sum of the areas of triangles on the other two sides.
Question 2.Prove that the area of the equilateral triangle described on the side of a square is half the area of the equilateral triangles described on its diagonal.
Answer:Need to prove that the area of the equilateral triangle described on the side of a square is half the area of the equilateral triangles described on its diagonal
⇒ Let us take a square with side ‘a’
⇒ Then the diagonal of square will be a√ 2
⇒ Area of equilateral triangle with side ‘a’ is ![](data:image/png;base64,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)
⇒ Area of equilateral triangle with side a√2 is ![](data:image/png;base64,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)
⇒ Ratio of two areas can be given as follows
⇒ ![](data:image/png;base64,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)
Hence proved
Question 3.D, E, F are mid points of sides BC, CA, AB to Δ ABC. Find the ratio of areas of ΔDEF and Δ ABC.
Answer:![](data:image/png;base64,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)
⇒ Given, D, E, F are mid points of BC, CA, AB
⇒ Need to find the ratios of Δ DEF and Δ ABC
⇒ DE || AF or DF || BE
⇒ similarly EF || AB or EF || DB
⇒ AFED is a parallelogram as both pair of opposite sides are parallel
⇒ By the property of parallelogram
⇒ ∠ DBE = ∠ DFE
Or ∠ DFE = ∠ ABC …………eq(1)
⇒ Similarly ∠ FEB = ∠ ACB …..eq(2)
⇒ In Δ DEF and Δ ABC from eq(1) and eq(2) we have
⇒ Δ DEF ∼ Δ CAB
⇒ ![](data:image/png;base64,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)
⇒ ar(Δ DEF) : ar(Δ ABC) = 1:4
Hence proved
Question 4.In Δ ABC, XY || AC and XY divides the triangle into two parts of equal area. Find the ratio of ![](data:image/png;base64,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)
Answer:Given, XY || AC
Need to find the ratio of AX:XB
![](data:image/png;base64,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)
⇒ ∠1 = ∠3 and ∠2 = ∠4 [corresponding angles]
⇒ Δ BXY ∼ Δ BAC
⇒
……….eq(1)
Also, we are given that
⇒ ar(Δ BXY) =
ar(Δ BAC)
⇒ ![](data:image/png;base64,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)
From (1) and (2)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEEAAAAfCAMAAACPrVDwAAAAAXNSR0IArs4c6QAAAJNQTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOgA6OgBmOjo6OjpmOjqQOmZmOma2OpDbZgAAZgA6ZjoAZma2ZpDbZrbbZrb/kDoAkDo6kGaQkJCQkLbbkNv/tmYAtmY6tmZmtpA6tpBmttv/tv//25A625Bm27Zm27a229u22/+22////7Zm/7aQ/9uQ/9u2//+2///bu+OnzQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABWklEQVRIS92V21KDMBCGNxWph1ooKniitTaggpi8/9O5u0DIOKPZKeONe1GSIftv9g/5CjA/TKqUWtwAdEsV20LFIkm79dbpqIH65AD8QxNBVFeZp1AmYJ9pboukWosEcLGeFLiLhOu2w1OwCV+hOz/Ax9kT7WG77JUk4e2BhibNUeDutiU7ZDEpYPNYHDM/s5yckOWDLU9HxzS6oC5fMBnPUasFtRMOzpJWC8v9+xUl+cXxm8FukRsEjPme8Mc+yro4dhNa4UWYFTvxDfqhjLmeVZ9Qglgcw3GSCFHIKAlQ4Q18y5Rak4zjJA4vhAr2vgGzeYSaPzXHSehWdRi0dk9cH2zoCG4TJ22RCyD1uscq7XCUmhgxcbKMeRIKLATDWda9DSMn29AVGqV19N43oVlg4mSK/TFyQ2E2D3yWLQpgOx4nowZhKaG15hNgA4/8truV9J8t1A6//wIjdBsCAgQenwAAAABJRU5ErkJggg==)
⇒ Now,
-1 = ![](data:image/png;base64,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)
⇒
= ![](data:image/png;base64,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)
∴ AX:XB = √2 -1 : 1
Question 5.Prove that the ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding medians.
Answer:Need to prove that the ratio of area of two similar triangles is equal to the square of the ratio of their corresponding medians
⇒ In case of two similar triangles ABC and PQR we have
⇒ ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
⇒ Let us assume AD and PM are the medians of these two triangles
Then
⇒
=
= ![](data:image/png;base64,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)
Hence, ar(Δ ABC) : ar(Δ PQR) = AD2 : PM2
Question 6.Δ ABC ∼ Δ DEF. BC = 3 cm EF = 4 cm and area of ΔABC = 54cm2. Determine the area of ΔDEF.
Answer:Given, BC = 3 cm EF = 4 cm
Also, area of Δ ABC = 54cm2
Need to find area of Δ DEF
⇒ since, Δ ABC ∼ Δ DEF
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒
= ar(Δ DEF)
⇒ ar(Δ DEF) = 96 cm2
Hence, the area of Δ DEF is 96cm2
Question 7.ABC is a triangle and PQ is a straight line meeting AB and P and AC in Q. If AP = 1 cm. and BP = 3 cm, AQ = 1.5 cm., CQ = 4.5 cm.
Prove that
(area of ΔAPQ) ![](data:image/png;base64,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)
Answer:Given, AP = 1cm, BP = 3cm and AQ = 1.5cm,CQ = 4.5cm
Need to prove (area of Δ APQ) =
area of Δ ABC
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPcAAACwCAYAAADT0fiHAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAADPdSURBVHhe7V0HYJRF9v+l994bSUgDQpMuTaqAeIj0agHl7GI/y1/PLvYTFRBEFAW8UxQRERBR6UpRkF4SIKSQRnpP/u/Nlmzq7ibbspnRkOy38015M2/KK79nX0MJMkkKSApYHQXsra5HskM6UaCalnRe1+1sbXTKLzO1PQpI5m57Y2aQFr/+XTJSckrx6sw4eDjbGaRMWYhlUUAyt2WNh0lacyGrFOt/v4LM/HLc1DsQo7v7mqReWYlpKSCZ27T0tojatvyZhY6BLriemPqr/RkY1c0XNvJ0bhFjY8hGSOY2JDXbQFllldXYciQHE3oHoGeUO+5acRKn0orQKdStDbReNlEfCkjm1odaVpD3cFIBcovKxa4d4uOEcF8ncUR/amK0FfROdkGTApK529l8+O5gJrILKrCBfjva2+JqcQW2Hc3BA2M7wF0K1qxqNkjmtqrhbL4zecWV2H8mD8HeTvj9bD6qSR8W6OmEP87nYdfJXIzt6d+OqGH9XZXMbf1jrO7htqPZKC6vwtcP90AoHcnZfKma/vnn8hNYsztDMreVzQXJ3FY2oE11p4IEaV/T3bpvjJdgbE4sIbejfyb2DcQ9H5/A8ZQidAmXgjVrmRKSua1lJLX0o7CsGiF0HJ81KKhBzoHxXpg9OASXyKhFMrf1TAjJ3NYzls32xNvVHm/PjW80j6+7A1mqxYojukzWQwHJ3NYzls32RBcjFVtdMrUTellDNyVzW8Moyj5ICjRCAcncVjgtzp46hs2btyA1IwthwQEYN24MYuK7WGFPZZeao4BkbiubH198tgpPvbUKqTYdYeMWgJrCI3hr5Qa8+vjtmDH7VivrreyOZO52Mge2bdmEe55fjtKuC+Ad3pUctqtI32WH1EtH6fkKBAUGY/joMe2EGrKbcue2kjlQUV6GFavWIj98PAI7dENNWYG6Z970+UruGCxduRpDh4+Anb2DlfRadkPu3O1gDmRmpOJkajFcgxOAyrK6Pa4qgzM9P5t+CFlX0hEUGtEOKCK7KHduq5kDZG5WTcdw1lU38M3mBzX0VRVZpUnHbasZci0dkcxtJSMdEByKrpGeOHrhGNyDOgJVFUpGZxtTJ5SmHUPnKE/4BQRbSY9lN7RRQDK3Ngq1ke8dHBxx57xbsWn+k8jz8IN7ZD+xS9N+jcJzu+Gf/iPue20R3bflkLeRIW11M+VIt5qEllNAiX8/RPSfjHP716Dk0i+odvaHTUkWbApTET10ClyjBlpOY2VLjE4BydxGJ7FpKjiWUogPtyTBs8t4vD59FHyKjuPi5SuICBuEJMTju9POWPzDOSyaFQs/siWXyfopIJnbCsaYQRiWbE3Bhcxi4b454/pucHHqQ44gAIGtEMppFVJKz2Dv6Vx8+ksqHhofKQERrWDctXVBMrc2Cln49ywcX/XrZew6dRXdO3hg9qAQ2NtUorCoWt1yL2db3Do0GMnE/OzT3TXCHdf38LPwnsnmtZYCkrlbS0Ezv7/1SDb+t/cK/D0dMG94KIEeOqKkvJaxuXllFdVIDHfHzIHBeP/HS/jo5xTEh7oiKsDFzK2X1RuTApK5jUldI5d9Nr0Yy35KQSmhrNw6LBS9oz1RWo+xuQm8u1dU1WDcNf44cbkImw9nYem2y3hhWgyBJEq9t5GHyWzFS+Y2G+lbV3FBaSU+pHv26fQijL8mAP8gHPJKumQ3BbdQRd850QWcEVfOZ5QQ4mkWHePdMWuw1Hu3biQs923J3JY7Ns22bM2udOw4noOudNy+ZWgInB1sxfG7ucQBCaLpKD53SAhe+y6J7uqpSIxwQ49ID4ukwvHjx+Ho6IjY2NhG21dRUYEffvgBFy9eFN9zYMNevXph8ODBFtkfUzdKMrepKW6A+nYcy8Xa3enwIuik2+g4HuHn3OCe3VQ1pbQADOrkjYmpgfhiVxqWbLskggH6uFmWeuzEiROYMmUKHnvssSaZ+8yZM3j00UcREREBNzc3VFZWwt3dXTK3cvAlcxuA2UxZRHJWCZb+dAl5JZWYPzwMA+K8te7Ymu1jnDR7slybMiAIJ1KLSD2Wh9U703D/mA4WoR7j3XjDhg147bXXcO7cOTg7OzdJXt7ZExISsHHjRrXNvAw3X0suydzNcGZeXh54svn7WwZYP++6S7elCKHYyK5+uJl02hxYgPXZ+qTyyhr4ezjg9utCcYkifv5vXwapxzwwItFHn2KMkvfy5cv49ttvce+99womLyoqarIe3rmDgoJEvtTUVIwdOxYdO5JdvUyCApK5m5gIzNT33HMPRowYgfnz51vEdFlHR/FtFMQvLtiV9NYhIvwPM3xLEkvVe0Z5YNq1wULi/hEdzxNCXBFGscPMmUJCQrBs2TJxzGambWonrq6uxsGDB7Fr1y7k5uYiOzsbixcvxltvvYUbbrjBnF2wmLolczcyFHzfe/HFF7F27VoMHTrUIgZrz+mrdHxOhasjG6SEomOQi8737MY6wJs9q8du7BUgonxu+ysby7en4EkKCOhEwjlzJScnJ/BPVVUVnUqaXriY6VlwNmbMGLH4spPMM888g1dffVU89/T0NFcXLKZeydz1hiIzMxPPPvssfH19MWrUKHEsN3dKzS0Taq/swkohGR/Sme7ZdLRubaok5nZ3ssUcUo+xzvx70n+z9Rrfx82dtN2d7ezssHDhwjrNnDt3LgFDbkZKSgq6dJGAkJK5681i3jWef/55MTlmz55tduZm9RUfm49cKBRMPZUYT8T40vei3QS38rE+hk4Bt9Bp4I2NyfiEbM87k3ot0cLDChUUFOC9997DzJkz1fdsPp7zDs5HepnknbvBHODjnGrV513b3MglX+/PwA+HMxEV6IzbSe3l6WLf4nt2U8fzcmLw67r4iFhhX+1NJycUVo/FwoPqMmdi1Zbm0Zw/88nK29tbHN13796NvXv34u2330Z5ebk4lg8YMADh4eHmbLbF1G3e0bMYMlhmQw6cy8env6aJONp8HGeBV327cUO0nE7ncKQdbwbZnp+h+zc7oXxORjJ3jzYvk3h4eAgmViWWpD/44IPiOD5s2DCxcz/xxBOYOHGiWIT79u0rGJyP7DLJndti50BGXjk+oB00/WoZZpKn14hEX7pnVzdpXtrajpRT2UFejphLi8ir3yZh3Z50YZ46KMG7tUW36H1m0EWLFtURjLHa66WXXlLvzGy5tm7dOhw7dgz2hDCTmJho9pNWizprpJfkzt0MYflIqE2wY4xxYRvxFT9fxsHz+egf54XptKPyzlTFW6wRE9+/OcTvlP5B+Gj7ZSyhu34sqd2Y6U2duL9RUVF1qmWDlq5dCY9dIzk4OKBnz56mbl6bqE8ydzPDZK779ncHruC7g5kI9+d7dhj83OxR0kJ9tj6zkBcyltNN6B0oDGV+OZGLj2mReeKmKNjZSu8xfWhpCXklczczCnfddZewgDJlOnqxACt2pFKVNZgzKFiopkorCLLYRKmCVGwernZCep50pUQsMonUhpv6BJioBbIaQ1FAMnczlBw5cqSh6KxTObmFFXTPTkFKdqk4Gl/f3R98FzZ12Gy2XksgMAd2D333h4tYueOyUI3xEV2mtkMBydwWMlbs0LHyl8vYR5ZoDLowi4RoLPRlO3BzJK53dHc/4VzyDUEzsXrspRlxcCELOZnaBgUkc1vIOG3+M1vgmwV6Owk3zkAvB6OovXTtrgrcgaGZ2Hptx/FckqCnCRmATG2DApK5LWCcTpLw6iOSTLM5KDNTTwJPaKlDiCG7w6q3cF9ncf9+5ZskofvuQtZr/WO9DFmNLMtIFJDMbSTC6loswxJ/SB5ZSZklQkp9A+GcVVSZ/p7dVHtZmNc/1hOT+gWQaWqaMIVlc1V/D9Orx3SlqcynoIBkbjPOBBaUsafXb3Tk7Ub+1HOGBAucM94xLSWxaozbOalfkIgiupdkAp+QNP+Rf0TCVgYVtJRharQdkrnNODw//Z2NL/dkwM/TEfPonh1KvtQlZZbD2CrSsGuoF8EwMfZacmYpvvnjCroRRvrYnhL73IzTR2vVkrm1ksg4Gc5nFAtUlWJSO80hpundsXFYYuPUrn+pZXQ870bmqDMHBmHxlktkQZeChDBXAbgok2VSQDK3GcalsLQK7zMscVqx2P3YQKSKzr6m1mfr03VuG6vHxvT0F8fzTeSpxkLA5yZ3hLOjdNTQh5amyiuZ21SU1qhnLamUfiFY4s5kGHLLkFDSHbccLsmUzWf1GEMos3ELq8e2EnoLW9DxZ5ksjwKSuU08Jr+SvfYXO9Ph4USwxKRi4pA+xeWmMy9tbXfZ97sD2byz99jr3yXjs99SST3mhmuiJKxRa2lr6Pclcxuaos2Ud5HMSpfQPZthieeRMchAcqc0pd24IbrK9nJ8/x6c4IOTvYsI+zxdyA4WzYqHNzm4yGQ5FJCjYaKxYKOUZaTPPk5xtNk3++Z+geBjroHQkkzUC0U1IjQwOYkx1hrLDfadySNQiVQ8MM4ysM9NSgwLrkwyt4kG58u9GfiRTExjg1yFxZdHK2CJTdTkZqthhxbGPue+XKBACV8RHBRL00d09bWE5sk2EAUkc5tgGvx+Nl/cTV2cyJXyuhCBO17Shu7ZTZGIvce6R7pjOmGf89GcAR64bxF0J5fJ/BSQzG3kMWCYpA+2XkROQYWwQBvayUfcs83j62XYznIfGB1mfC9/HCf7+G0UK3w5gTs8N6WjBHcwLKlbVJpk7haRTbeX2IyUd7PDSQUYTMH3pgzgcLk2dGe1BtZW0IAhoVxJlXcLGeIwuMPmP7PE8ZwhmGUyLwUkcxuR/t+Smeb3hzIRSeouduP0pqicluDtZeguc584AspsivX99qYLIjQwe49ZOva5oelgaeVJ5jbSiBxOzhcA/4w9NocmfadQd6u4ZzdFLrZeG97FV1iv/Zfs5Tk08MvTY0WYYZnMQwFJeSPQPaugHO//eAmpueWYcW0QRhGiCUuXrTmpwB1YuCawz0+Ssc7uNNwzOsKau23RfZPMbeDhYcCFj39OxR8ES9yPQA2mE8ghu0aWk4+2tSeWMQQRggwHKnyZwB3+S+o/Nk9lIaJMpqeAZG4D0pwFZewO+c0fmQj1IVhiin8dQKAG1qD20pVMZXT/Zg83gX1OnmPsXBJHuv0QH/OGBta1/daUTzK3nqN5hiyyllGoW/bsYqwCFnyzQQqDCY7q5of9ZK3Fqq7ZQ8JFQHtrFKA1RzK2XuMj+gTydDtJ4Io7juXgY0JP/ddN0bC3k9jnek63VmWXzK0n+RgOaR0JjKaTX3Mo70Y0mS9kleKBVafwyoxYMak9SYg0uqufEi7JetReupKKwR3cacFjP/XzpB77/pAiNPDEvoG6FtFkvnQKs3Q4KR9FtLh2CnMT5crUOAUkc+s9M2wQ4eeEB8mOuoNfrSXWI6tPCxXQmge6CXxvlpLzDtZeEx/P4wn7fBZpCt4l9RjDNncOcxd46C1NDNDIUnh3svRzpbjil3PKhf3A0zdHw8/doaXFWu17krlbOLT1o+vwLn70YiHs6KzOYXZVx/YWFt/mX+NljaOXjCJbc0Z3ZezzpXT/fml6DNyIOfVNHPnkpa/P41HCbuMTgCvhpx+jkMMPfXYKz6w7i/fndZJWcfWIKplbz1nGTF1eUYMjxMiFpYpAgZfIlZOx0MYRcimD9meSqSkZorX7pFKPidDA6SX49UQO0ckN84brh33OYYvZVXYmaR7uGFH7bt8YT7xzSzxu/fAYdlPY4aGdpVRec9JJ5taTBR3sbJFTVIEXv06iu7UdBYevIaFZDYZ28cYj4zuIQAI8qSVvKwjL6rEwAn7k+OIcGvjznWki9pg+2OcsmMuhUEtjejQEZGSQCAa82HM6TzK33Ln15OZ62Vlf7Uuuji9MjSFhjptAUXFxsBPIpXzPvEo45DLVpQDThZl5EvmwryKrPb43Rwe6IJBQX3VJHKucU2AjoYRZAs9Xoiv5ijwy1VJA7tx6zgZWfTkTtjgD87Mr52myxppGVmiMrsKwxKwek7t2XaKqsM/5rszmqXtO5Qrs88cnRAl6aUsuhNvGWommwhgzgmyAlw4FaavIyr6XzK3ngPKRm4+axaTL3kEghwG0izPAYRlNMF0mqp7VWU12jqLiQyrCW+l4nkzqxG8pBjl7j3GEFW0pLsQVtsTfRy8UIJHUX5z4hMSOOByxhW0PxnSXIBH16SiZW9vMqvd9sLejOGLak2Tt2nhvsFDHjo6GbHYqU/MUKKVFkRFfOR7a+1suUhxyVo+5iSN6c4mP3cMJmmrptssYQkKzMPr8HVkCsrS8nOge6OnQ6H28vY+HZG49ZgBLxkOJuZ8iveo5CirQnXYe3j3YaEMm7RTgKw2rx1gwxkKyTaTe4thjz02J0Roa+KEbOhCc8knMWnwUU8m0tYCMWD7fnQ4Gw3iCjvcydllD+kvm1j4n1Tl2kbqFVTLsxjiRLNGCwh3ljq0H/TirUI9pYJ9vO5otpOccqqi5FEDCt4/v6iJUjj+TSWsZLRILx0UQzLILvib8NoZ5um+s9EDTpKFkbm2Ts5qk37b2yCgCTaBUHKF73ywC4e8T46WIEqLtffl9Awqw9Jyt+xj7fNGGZKEe68rY59HNY597ONsLPTf/MAIMX404jUj0wdWiSjEWUqxWS27J3I0wX3VVBdZ+sRrf/bgDBYVlCA2kOF5+vXDGvi9G9wgk8IUQmkQEl9SOzUtbu2axc80gkllwKKUvdqWJnfeVWXE6m5GqGJvbwbs6/8hUlwKSuevNiJLiQjzxr6ewdNNpVIQNhZ2rD6pOpgPJ69Ctx3HMnPcKfD2dUFRCVmgytZgCCuxzGwX2Od2/958rwJp9uVgw1Bd5Wam0BdsiMCS8xeXLFyW0cYM5sHTJEizenAyvYU/ByZmkuOJY7oCymGuRtPcNnN2/Cd1jZsKxyrHxI7n6nK74Q/xb+4+WZ3UP+QocxfrPrOciwOg07PBx27AIZBRewsefrsOmxbuQnVcqrjyJkb64/54F6DdgkOTVFlBA7twaRCsuyMPqb3+GY+JMYmwn1FQUK76tKoezRwByIm/Ayi++QlznrrB3ckUNbT+2pIC1obuf+E07kQ3tOIrfih9b+q7uM+X34nnD/OIZl1evHHUZpO/lq6bwJVe1XYPf9XrW5EJUu6g0tjjVqbaRBUjX7zlfWaUNupEJaVj2Zmzctg5HO0+GU0CcsNk/fPYItt36CD567RFMuHlqC6Z3+35FMrfG+KelXUZOqR0cPQJRU1nXnLGmqgyuAdE4+vtafLFyKVy8/Ii52XCllpEVzGyrZHj6TQyqYH7+rWBk1feKz3a1i4M6r0YZ/IyZXFVmnfIVC4CtLZVBenYuy47s3hWLjK34W9RnR9+LBYPrqi1bvKfZJo187K6qqtOWyrVRl8sLC60u3A5hiseSB0VeFmSJ3+Jx3QVOkFhY7mmKu7jNZO1Hft9HDh3E5o3r4TbwAbiFJtJiWiZe8AyJwZUTvnj61Q/Rt19/hIR1aN/cqmfvJXNrEMzL0xNOduT4UUGTy8aDtkYN3DOa5FWlBfD2dEVUTDwcnN0IkL9SCNVqKB/vNPxTTQzPTC/+5mfi72pUkoRY8VmRv/Y75TvifX5X8z2WxtPWyP+LM7qiDv6TnytYpdbeVcU8CgZTMp/m4sPspfFZMGcTP2rGVX0vThPKE4rqtMKLk4qRlQuHgvEVzF27kNFnXiDEYkbPaSGxUy4ubm5u2PjtV0h36grP4M6oKSe1hOpMUl0Bv/jBOPHLbuzfsxMTp87Wc3q37+ySuTXG3z84DEO7heKTv3fCJmAOWVwUCYazIVVYDU3MwuNbMXnKcMy/50GUllUSc1epmVnFrIK5mXnpt/qZimmVjF+t+bl+Xo33FOUoFgh+R7VwKOqoIZ0xLwaKNogfMvFUlK3KzwsHRTdRfV8vXzW1n+sQ5ZAhTnWNRlkabVWVz/kqK3lBq20LP+PVRi1hUKxFyqRcmOp91jz0Ozk44Mhff8HGbSw95sVU845B1x47e1Q7+SLjypX2zakt6L1k7jpEs8EdC+7B1zPvQ/afLnCPuQ52jk6oLLyK/KPf49rQq5hz6+0iyiUzFx9t7WgXUidxTFUlxY6q++c6m3Dti/Xv03XnvpoZNIVvimuwapdXsl2dz8rdXxwBFKcDxZ+anxV/i+goGqcFzXJVpxV+xkzOi4V6oRGLEX3mBUdjMeC/xaKoXKycnRyRc/Uq/tqTSq3g4z1TTNlJOgFUk0DTpjQbAX4N3T1bMN/b1SuSuTWGmzeh/bnhCBt6B/xOfo/S48dQWG4LL4dKTOgdjIUPvomwyCiUlqju4wqmME9S3Fm1J8UKI+7IDVL9BUgzgzJ/3V/KDI0+1N6URnI4O9vA3t4OG3bcj+zUE/CO6EryDr5zU7J3Rt7ZXbjGvwDXDhnWovLb80uSuTVGf9PhTKzdmYwOidfioQfHw6viIrKzc+HrF4COCV3h4Oigwdjmnja6Liz1j8bmbnfd+osqbdGlW0889s8ZePLN95F59Wa4BifQolmF0tS/UXVqA2Y+v1AK01owbJK5lUQ7QThfHLSPvbum9fHCtZ2DUAUKZscqJ9rRK8iiqrRUAgK0YI41+wrf+Zm2t955D/z8/bH2y69w9sJeONHM9HR1xNVR83DcfjA5iJSDPfJk0p0CkrmJVuwT/MGWS7hAfsYMKDC2pz8JzCqE/bJMxqcA38H5/n7ztOkYOWY8rqSlgOGsXH3CsHhHNrb/mUrBDWzwzKSOQtouk24UaPfMzXfmTwmSmD2+ekR6iEiV7LXEzg0ymY4CLGgrKion4yFXRMd1EiI1e+LjWf2rcC4lB5sOZ6E7jY8hsM9N1yvz1tTumZtdDv+7LwN+hKhyO6FyctgbBjmUyTwUYFUb/S9SOTF3AqGwsBcehwZeQTHYGNwhIVSBxiJT8xRo18x9PqNEYGkzM99Cwet6k8sheyvJZBkU4FNVKflts0sngzt8TYvwUgJXfGlGXIuwzy2jV6ZrRbtlbg4a8MHWSwJ/6wa6Y0/oFUB3bDYOMR3xZU3aKcBGPE4ESMmhkE8TuOIvx69iDSGw3KmBX669lPaZo90y9xqKHc1B6hIpxA2DBrhQeJr2FrSvrUx5BqTk65IK+3wtjV0igTsMJH9wmZqmQLtk7l8ItfSLnenwoLA/tw0LFVA97SnMbltkCIF9HueFm5XY58t+ugxGReUQyTI1ToF2x9wXKfQPo37kE874fDraDaAJIyXjls8erJXkI/rNpKo8TVepnSdysZIEbI/cGClDAzcxfO2KuUtJcPbhlhQcJ0jckRRLmycK204L+2mZLJ4CjDLr7eYgrlFJFBr4G4I31hX73OI7Z4QGtivm/nJvOrYdzUIsHef4/sYxpOU92wizyohFsjajM6nCODTwe5sJ+/zny4in8eSwyTLVpUC7Ye49p6/i09/SRHSQW68LoclA92wK/yNT26IAH7LYRPj67v5gk+GNBzLJei0FL0yLgTONrUy1FGgXzJ2WWyb0o7kUKXIu6bOHdPIBH9HlYbxtsgKbBauxz+n+vZ20Hl33eYjTmEztiLk53Myy7Sn480IhMbU3pvQPJE8QgiWW9+w2zQcsBI3wdcacIaF44/tkrN6ZSsEN3IQhkkwKClj9zr2eolFsOpSFKFJ33XZdKLzc7MWuLVPbpwDLSwbTgn0qPQCryT9gydYUvD47Hr7uVj+tdRo8q6bCoaQCfELxoB0IQJAlrAmh7lKfrdO0aBuZFKcvG0wm3fcpun//fjaP4n9fxkPjI3UEsmgb/WxpK62WuTPzK4QbZ3pemYgqOYKiRDJOtkzWRQEF9rkjCUlDcTGzFOt/z0BXCtB4fXfTwzLt3bsXe/bswQMPPAAHwoarn64QDtyLL76IpKQkAR7Jrq7z58/HpEmTjDIoVsncLHBZQffsP87niXC7068NFn7AkrmNMofMXiirx1jfPY0W8SUkOF1OoBudyHuM45GZKmVnZ+Pee+8VTPvggw82Wu2hQ4ewbds2zJ07lyCdnQXYZFhYmNGaaJXMvZFCw26gn3ASuNw+LIzuYI7S28toU8j8BfPpnEMDj+/lL0ITbf4rS1ghPj+lIxzI6cQUaQlFqsnJyUGPHj2UMNQNaz1+/DhGjRqFp59+2hRNsj6B2t+XCrGcDBv4Njab/IC7hruT26B04zTJbDJjJRwa2IVANti45RxZr205ki3ip8+g3dzYaf369Th//jwefvhhbN++XaC9NpbOnTuH9PR0PPTQQ8jIyMDkyZPFj7FSi3buX3/9FevWrSPUSnvMnj0bAwYMMFb79Co3p6gC79M9O4Xsx6dQgHa+d5UTtK7UeulFxjabmdVjHQNdSD0Wgte/S8YqkqB3oeM5I7gYK124cAGffPKJ+i5dUVGhCAhRL5WWluLEiRPge3f37t3FAvDYY48hLy8P8+bNM0rz9GZuXqUeeeQRjBs3TggE7rjjDrz77rviuGHOxDv1KpKM7yNLtF4Ue4qFaPYkJZf3bHOOimnr5jnA6jG2Zzh5OYD8vik0MHmPvTozFl6uek91rY0vKyvDm2++idGjR6Nnz544ffq0wLF3dGzoqcYb4ZNPPonIyEjEx8eLsuPi4vDRRx9h6tSp8PAw/AKkd49//vln3HbbbXjuuedEA1nax7u4uZn7B8LY+oqQOgK8HOmeHYogQsqUcEla56fVZWDvMVrTMXUAqcdSC8Fmx5/vTMO9YyIM3teNGzeKuc88wEy+b98+nDx5Em+88YbY9Hx8fOrUOWzYsDpS9H79+uGzzz4Tu7dFMPcLL7wAFxcXITS4fPky8vPzMXDgQIMTTp8CWYiynKTjLCWfOSgEPaM9pEOIPgS0srwsXAvwdBTQWZeyk/HlXlKPRbjjui51ma213WbmHT9+PFJSUgQv8E9hYaHYwfkYrpkyMzNx33334amnnkLv3r3FVwcOHICrqyu8vb1b25RG39d75/b19RUFffrpp3j55ZfF6vSf//zHKI3TpVAVXBLjof2jTwDGX+MvHAvkPVsX6llvHr5/9yJTVN7Bl5FjCUvP2XMszNfJYJ0eOXIk+EeV+O69evVqfPDBB+JonpubKxi4b9++8CdMduaVBQsWiLs2C9QWL16M559/Hu7u7gZrk2ZBejO36uX+/fvjrbfewpo1a/DSSy+Jv3lHN2Xi0wPDEv964qowXJhD0nEJS2zKEbDcuvh4zie5Gwkbj73Hfibnko9Ji/LkzdHCYtEYKSIiAswXfL/mlJaWhhUrViA4OBjdunUTPPLOO++A1WbM0GzQMnPmTGM0RZSpF3MzM7GynhvWqVMn8cPCAbawOXPmjJACmjJtP5aLdXvShS3xPNJn86os79mmHAHLrotPcOyzP5ek5+dJPfY9hYtiYxeGajJG4jv14MGDhSELp4SEBKxcuVIYrHDy8vLCv//9bzz++ONiZ1ctAsZoi97MzdJxNq0bMmQI7r77btEmFgZwaszczliN5nIZiWMZWSMVk0/2nUPC0CeGYYmlE6cxad4Wy+bjeVyIm1CPvbMpGSt3KKzXGP/c0ImZVZNhWXLO8cfrJ75nmyLptXNzwxMTE7Fo0SK1IOCVV17B8OHD1eJ9UzS6uEwBS3wqrUg47U+kuzYbMSgC1MskKVBLAZ4R5WTExL4FfDxfv/+KwKp/ZUas1WOf68XcTDIWBvBq9PHHHwtFPOu72eKmTpxqI8+utYRbvePvXHQiLy+WiDIChwQ5NDLR23DxZMcERwcbYftwhjaEX4/nYs2uNNw5MrwN90p70/Vmbr4rLFy4UOj2mLn5HmHKtPPkVXLMT4cb3aXYEygqQMISm5L+bbWucjqec5TQW2kzeOWbZKwhWU03Uo8NsGLsc72ZWzW4xlC6a5s4l8isdAkdx/NKKiiuV6gApZc7tjaqye9VFGCZTB/yEpxE6jGWnC+h43l0kBuCvBq6Z1oD1VrM3KbufAX57S4jXSU7hgyj+9OkPhKW2NRj0NbrY5kMx1pnGc1Jun/vJBUqC9ieuCkK1hgZuM0w9xqCJd58JAsxZIjA5qUcLUTCErd1djN9+xn7nO3M+XieTPHYNxy4QtZrbvhH7wDTN8bINbYJ5v79XB5hZKXB2UGhs4wJ4nu2RFUx8tyw2uIZQ4/DALM76Ps/XhS7N6vGrA373OKZOz2vnKKEXEJWQYXQVbJ9cBnZDsskKdBSCvDsqSQR+pgefuRcUkS7d6YwT31hegxcrQj73KKZmwdgOQk9DhLQ4eAEL0wlH21b8pXl5zJJCrSGAmya6szgDoNIPZZeLMxTu+9TqFatJVk0c6///Qo2HspEpL8zmZeGw4fMTOVx3Fqmnvn7wTKbcMJZ4/v3q98mYTVFpOlMoYH7djStetdYlLBY5v4zWQFLbEd2unMIlphNBovLJVySsSZCey2XY39fG8+hgYOEExIfz6NmUmhgz7avHrNI5s6hsD9sXppKYYCmEXLpqK5+dM+WR/H2yoDG7DfDndWQn8ckciY5SeAO+8+Q8Pa3VCwk7PO2rh6zOObmGMwcufF3InKfGC8h0bSTsMTGnN/tvmwGd/B1dxD37QuEff41RakxF/a5IQfD4pib79jrKe5ysI8T5pEVWoCHA0robiSTpIAxKcDY54yWykg+72+5KCKHxge7ISrQdNjnhu6fRTH38ZRCsWvz7s1qL/a9lYYqhh5yWV5jFBDY52TgMq6nH1mvFYIx+T4i6K5nCfucpeptMVkMc+eXVApYYj4WTaT7z9ge/uSqJ+GS2uKkaqttZnAHRvJh7PMzBNu1lbDPu3ZwI3VZ2wwNbBHMzUYFKymA2+7TeegR5SGCCTAUjhSitVU2abvtZkck9jRkS0jGPv+ULCMTwz3QI9I4OGfGpJRFMPfWv7LxX0Ko9Kf79TyyGw/1dpJqL2OOuiy7WQowg6uwzxk7YCkh/rw6Kw7eRsA+N+ZQmJ25z6QVC3RKDh4wf3iYCChQQsINmSQFzEUBBfa5DSGnBuE0gTvsPUMYAqQeu39sB3M1qUX1mpW52SjlQ1oVz5L53ziCJOZAbmwWKNGSWjSW8iUDUoBdjNmQhdVjF7NK8V8KeNGNwhIN62xY7HMDNrlBUWZlbo7l9AshmHahYH2MqsJeXxJ8wZjDLcvWhwKsqbmGZEBsSLXsp0sCkDMuyLDY5/q0R9+8ZmPuX47n4EuCuvF2c8BtdM+OoHC70rxU3+GT+Y1JAT5BVpEE/UY6UTK4w7aj2SL299OTjId9bsj+mIW5L2SV4MOtKSgopkCCI0MwIM5Lxs825KjKsgxGgQq6JjJe31zybzh/pRg//JkpwB2m0H3c0pPJmbuE7tlLibF5JRzV3Rc39Q0QRissxJBJUsASKcDgDgwQMotUtO9suiBCA3emq2QieZBZcjI5c7NqgY0D4kNccRvds92c5D3bkieIbJuCAmxQNZyw+04SuANHk2X12MvTY+FpweoxkzK3CKe6Kx2uxNAsQIsmY4FiCZck+acNUICDXjja22AGCdfOppVgF0FsryWZ0T9HWS72ucmYOzWnDB+QeWluUYVQLwzu5CPD/7SBSS2bWEsB3r1DCPv8Frp/v/JtqYhTx6GBByV4WySZTMLc7FK3jIzwj1wkWGLCQJvcP1CE/qmWCm2LnBSyUU1TgD0UOS4dz2F2cuLQRHwfDyarSktLJmHu/9EdZdOhLEQHuojjuDfBEks3TkubCrI9ulCA9yMWAN9EUMgsFObQRCsJMejxCVGwtzB0B6Mz94Fz+UK6yLGamLHjSJAmcdB0mUYyj6VSgF1DGTefr5eMfb7xIIUGpuO5pWGfG5W5L9M9ezE5vl/JL8MMcpsbTkfyChlm11LnrGyXHhRgS0rGPmdwh3d/uCjCEzHOXxwFzbCU1Grm/uloDnl0pZNNOGFR0X8MiTSQAOfmDA7FoeR8HCagQzZSYSkjbMjiR96zLWXsZTtaQQE2y2D789HdCNyB1GPfEnoQgyu+OI2wz0kb1NpUWFqFlJxSAeXdgdB/WwIY0Wrm3n82D3vph5mXGZsdP9gH9nxGKe4eHY4hCT7CmsdXwhK3drzl+xZGAZ7rTva2Yu6zdyP7SbDrMptTtyaxjIqvsjmFlWLDDCXIsX9NiEa/WE+9im01czN6JEsPn745Wl3xdeQ58/gXp7FgZBgeuTGSVjJbec/Wa1hk5rZCAQYUYexzBnd4bUMyhZcmcAe6f/clnmhJWkUb45vfJ+O+68Mxkk4FLLxbRe6mC5Yfw2f3diWcNw+di201c/NRmxlcM/FxxdXRVij9g0gveLWoUucGyYySAm2NAnz/ZuzziX0D8dlvrB67hI6B8fAj8BF9UvrVcgHMuGBEGO4aHaF+9aVpscgnHmJVskmZ24Uwp1gifu/Kk0IVUES243+czafGhQndXzbF+OJrNl0dZJIUsEoKsL1GTY2N0H1z7LF9BMv9CcGGPfKPKN77dE4Hz+cLoBJeJDQTX3dfnxOvczmqjK3euVmw4EA7NIdF5fuHR5UdwkkAcJoAGLIpuIALCRcqiuXOrffIyBfaFAVYPebjpggNzF6PHAqrGx2hOdigrimDgl6yr4UPYajXT/xc39Rq5maPmd7RnnhlRixF4iyHvZ2t8IGd8OafYCeR+8ZEoKBE32bJ/JICbY8CDO7AaKkiNDCZWjM0MqvLogJ0wz63I/7lKy7fsw2RWs3cqnMHR948RNE4GRqWBWq80vBKxImP5FIDZojhkmVYMgV4jrP9OcNy8/GcrTIF9vlk3bDPOeBlKcmrmG/4JKyZOAppYUkVJvQJ0JkErWZuXmXYDO/Hv3LgTgzN2M8LPz1Fcb7KMZ5w0dibhhcife4eOrdeZpQUsDAK8HxXhAYOEaGBt/yZTX7f7gTXTXYeWlIvOgHz0f7zXal4YWqsOndhaSWe+985AR5qUubuEeWOfaTnXk4rFHOwDf3n5WqHN+fEoj8Zr+TSvZtUdZK7tY2s/N5qKMBIvpEUhuiWoWFYtCEJn+1MRReyXrsmunk1lieZtD5ACKtPrj0jNsnru/sL7H4WzvFx/aHx+qGvtmrnZvPS7qTT++SuLmQ7bgtbBXer8Z05WicfU6Sk3GrmreyIDhTgvYzVY4MTvHCCHEzW7EkT3mOvEfY578zNpZvo2M12IR/9dBm/nTgnjFi6R3jg5XvjRLAEfVKLmZtBFt4myBk2vZtGFmiT+wcRE9vQMbyaLGsqRNwlqQLTZyhkXmuiAO+09qRFmnotY58XC/WYCvtc22bHJq0jKWw1h9hiRzPe0VuS9HqrKC9bMLCLhy/W7MvFDjK3Y0kgO6zzUYR3aZEUG7jcsVsyIvIdq6EA80SApyN5Q4bgYnYJ2KyU1WPDE7VjnzNTtzbCiU7MffD3PVi6fBUOnEojnbYtosMDkeIxBP4deuPOkcFqN05tK5LVjJrsiKSAjhRgQNDuFMxg+oBgfEiWa0I9FuZFIbOMHzlUK3Nv3bwRtzz8GjI8BsM1YphAUDlw6jBw4T08+tjDGJyYiLJSEprJJCkgKdCAAgL7nI7oNxD2+Ql2LjlVhLfX/oboyr+QfDEFXp4eGHv9SPTpP8jg1GuWuVMvJmHhs28jK3I2AuMHApWlogEe4YnI8w7H9vWf4J+TBiM8sgPKyhTxvfg4XlcFr10hr86hPauoo9ls9b7UXmRtDn3boTka2utR5m4io87v6zAF9KFPw+Ja35LWl9BIJ3UoVIcsOlDP8FnYBsTVyR7zR3XEoV1L8J9Vq2Hrl4BKNwJXLE3HO188jcduGYUn/vUkbGz1t0RrqsXNMve27T/hXHkYvKP7oKYsv9YSpaoCHjED8fcvu/D+6o3oM3oGysvIYMWmRggA+F6u0muLw4fGMz66ix/+T303V/zNSYFUQ5/pRf5ThVyjKFP5jsadXjxTnnA066r9u2FdtWUq2qpalFRtULVLpdpTNkkhR1AKFNR5+JmyPfyHqt8qGqjLVNNB0U91Ocr3lU8U76tppMzL5WoYCjRmM6B+VufL2r7VmQDKPLqX05gmU9Hp5spovE1NvKPqeL2Z2kh31Dkaq7vpNmkhYH0OaWSlaPhI+3IictA/LqQevnrkd5zb9SXsus6Fe3R/2NbQlk6Ttyg3Dc+seA8B/n6Yv+Bug60uzTL3+eRLqHYNJnoxEoNGR6hRdnYOsPMMwdqfjmFnxQXUVLL9eLVycioJycYr9SZS7SRtODnqTGAVkyu7Wv8+r1mOmiE09OmaDKmaTvXbojkRBOOq6lQzq4rlNBlYJSxUMo6SEVUMq7lAqBcNVdmaC4mKicVv5aKgwSyaC2TtgqhRN+XlBUzxnep9xSKgXryUC58in6K9YvHlz/SH+K0uQ1VO/QVUcxGuXZj5xfr1q8tUjrlqoVYs+LXvNv53bT80y6m/AdTdFGrLtFUu8Jr51RuFBk0Y/EDQQdl/JfnFs9o5oNEW1fyttwnUzq/aeVy74GuMp+i3LZzJwGvpis+Q4toHAbQx1pQXKriekrtvKMoSZ2L52q8xecpUePv6G4TBm2XuAD8f2JSScYoYRo1EvaiuqUJlcRbGDeqJvsPC6FheJu7jvAZwk9V/8zPxWfFcZTcrgOb4OUf1VH9PfwtEF+X7nF984N+KfJy4DPU7olxFvfXz1rahXplKF1WBwCoKVJTB7yurq9MuVTtFVlV/lO2s2xYFoquqneIv5QfVCq5JRtV6WfuGxre1ryqKqLdJqOuo81yj7sY2FSWtRHGNfV+/Gs22K5tWrzrxtH77NdvbXNvrzCmNttWhUb2+1212E51oSK5aQ6omXmmwmHK9asbWWOT5sSbDqxZL8bx2wRevK7+ztXcgXsnD8f3n4RE3hdrCMqrahtRUlcLVNxyX0p2RnHQWPU3B3NePHoWg5d8gKzMJnsEx1CbFndvG0QVXL/yNcFsyk7v/RURE6R43qXZSKaeExiSuP+nU3ddgmKbz1CGXmiE0x7LO4UOjoNp6asvQfKY5gRuWoZiKjbW1duFQLgrKl9WLiHIlUS8YTAvNxbDO35oLnoJoYtERZWgsTJp1KDlL1KdkHvXiq7GIcllkliCSwn1RuWCq89RdQFWLVoNyVW0Si6+yHHXdivar3uES67+v8pdgHbHifcUc0cQLaEAr1QJUbxPR3GjEGqsqS0kT9WajHIO67VXWqxof5SLd2Lip+iFop9x0VPNB1EE/Nnb2qCSd9UUCCc2tYtmUcnXQWMVsxO5UKXZ5Q6Vmd+74zt3wf/dMw8NvL0VW4VS4BBCDUyNKzh+Ee/I3eOGZW4ixa21gdWlUY8dp7e81JIb2d2QOSYGWUUDbJiC+5ynZYGPS2I01FlTewvmdF1O64rXNB+AS0YsOw+wCRozOVyp7VxSkH6Ydm2KSxervt91UL7WqwhbcfR+Cg4LwwadfI+nsVnHUiA12w33vLMS4Gye2jHryLUkBC6ZAA+FlfYFPk21vfhO688552Lh9AY4dWA/fHjfAztGdrn9VyLvwJ5xOr8X9i+6Du6e3wSijlbm5pgmTpmLCxJvpPnCOBBG26BDdkZ4aTmRvsN7IgiQFLJgCHWMT8NmSRXj0/17G7h2HUeMeApTkIdI5C//37O2YMn2WQVuvE3OLGm3tERWTYNDKZWGSAu2NAr36DsDG/32O/Xt24nxSMvx8fTBg4GCEhEcanBS6M7fBq5YFSgq0Twq4eXhhxJgbMcLI3ZfMbWQCy+IlBcxFAcnc5qK8rFdSwMgUkMxtZALL4iUFzEWB/wfv9+0k89yDhAAAAABJRU5ErkJggg==)
⇒ it is evident that Δ ABC ∼ Δ APQ we know that
⇒
= ![](data:image/png;base64,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)
=
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAAeCAMAAADEgrRGAAAAAXNSR0IArs4c6QAAAFdQTFRFAAAAAAAAADqQAGaQAGa2OgAAOjo6OmZmOmaQOma2OpC2OpDbZgA6ZjoAZjo6kDoAkNv/tmYAtmY625A625Bm27Zm29u229vb2////9uQ/9u2//+2///bmbpMdwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAdklEQVQoU2NgwApkhFiQxUXYeVH4DAzClPFlBJklkCwQZgQCbuxOoVAUZDISoNA0YrTDw06UjwPoSVjYyQhxQb0MCTtECIJZMvwcfGwg9VAZSVZOcTF+hEppHgGgDBNIJTjsZPg5JWRA8rCwk+JlZOREDlSwWwHrIAThsjMWcwAAAABJRU5ErkJggg==)
⇒ Area of Δ APQ =
Area of Δ ABC
Hence proved
Question 8.The areas of two similar triangles are 81cm2 and 49cm2 respectively. If the attitude of the bigger triangle is 4.5 cm. Find the corresponding attitude of the smaller triangle.
Answer:Given, area of two similar triangles as 81cm2 and 49cm2
Altitude of the bigger triangle is 4.5cm
Need to find out the corresponding altitude of the smaller triangle
⇒ Δ ABC = Δ DEF
⇒ AP and DQ are corresponding altitude of triangle
⇒
= ![](data:image/png;base64,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)
⇒
= ![](data:image/png;base64,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)
⇒
= ![](data:image/png;base64,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)
⇒
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABwAAAAsCAMAAABBqSgLAAAAAXNSR0IArs4c6QAAAIdQTFRFAAAAAAAAAAA6AABmADo6ADqQAGaQAGa2OgAAOgA6OgBmOjoAOjpmOjqQOmZmOmaQOma2OpDbZgAAZgA6ZgBmZjoAZjpmZmYAZpDbZrbbZrb/kDoAkDo6kGaQkNv/tmYAtmY6ttuQtv//25A625Bm27Zm29uQ2////7Zm/9uQ/9u2//+2///bFG/7MgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABM0lEQVQ4T8VT2VLDMAy00xbMVZIUA65LDQ0l8fH/34ckJySh2DMcM+ghM9FKlrRaMfYXZnjBZzZ91GzTJYLOgH6zT2f6u0MatJc58KZNZ3YrAJ3k/BaDwvP5NLSBv6Ar5uqKsde1nIPg8zWMY8iNsaMZAIMuW4cBJyD6sGZJjc0yiSAvFWSfPksEWTGAwZxNJosEHUUcpcH1jP38kqAUf91s6E9RDXCQNCSIpTT0YwWF3T8pCOe0AvdqJS8eP8YeCAq7K9xrrSAKZVbS4nuCGoVL7xawP8OO1w8FDkEEgbckRdiLPXOaiqBgI0H+/hC0empx2WtJL/ItCCdKjsjFYAjc0LdetSNBwyU62Rfi1XhibwKbCXqp4iRBL17SJ9bxZebETN/El1KwIqegnoOMjL4FvQNvPxuAy4SMOAAAAABJRU5ErkJggg==)
⇒
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAeCAMAAADXVfSyAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6AABmADo6AGa2OgAAOgA6OmZmOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDoAkDo6kDqQkNv/tmYAtv//25A62////7Zm/9uQ/9u2//+2///bhMG7gwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAYUlEQVQYV4WOyQ6AIBBDi4oLiojiwvb/v6lA5GBi6KXT5E1ngFeGk2oG3Chg2gW6PoCdwXQKVlLgJKTnLNJuUsFsjF42IpeUB5JUBj+El2GNwm8X/BrvQw/Rcny+/I+p7wYbZQQtDZbUMwAAAABJRU5ErkJggg==)
⇒
= DQ
⇒ DQ = 3.5cm
Hence, the altitude of similar triangle is 3.5cm
Equilateral triangles are drawn on the three sides of a right angled triangle. Show that the area of the triangle on the hypotenuse is equal to the sum of the areas of triangles on the other two sides.
Answer:
Given, right angled triangle ABC with AC as hypotenuse
⇒ Let AB = a, BC = b, AC = c
⇒ we have a2 + b2 = c2 …………(1)
⇒ We know that area of equilateral triangle =
⇒ Area of ACD =
⇒ Area of BCF =
⇒ Area of AEB =
⇒ Area of AEB + Area of BCF =
⇒ From eq(1) we have a2 + b2 = c2
⇒ Area of AEB + Area of BCF = = Area of ACD
Hence, the area of the triangle on the hypotenuse is equal to the sum of the areas of triangles on the other two sides.
Question 2.
Prove that the area of the equilateral triangle described on the side of a square is half the area of the equilateral triangles described on its diagonal.
Answer:
Need to prove that the area of the equilateral triangle described on the side of a square is half the area of the equilateral triangles described on its diagonal
⇒ Let us take a square with side ‘a’
⇒ Then the diagonal of square will be a√ 2
⇒ Area of equilateral triangle with side ‘a’ is
⇒ Area of equilateral triangle with side a√2 is
⇒ Ratio of two areas can be given as follows
⇒
Hence proved
Question 3.
D, E, F are mid points of sides BC, CA, AB to Δ ABC. Find the ratio of areas of ΔDEF and Δ ABC.
Answer:
⇒ Given, D, E, F are mid points of BC, CA, AB
⇒ Need to find the ratios of Δ DEF and Δ ABC
⇒ DE || AF or DF || BE
⇒ similarly EF || AB or EF || DB
⇒ AFED is a parallelogram as both pair of opposite sides are parallel
⇒ By the property of parallelogram
⇒ ∠ DBE = ∠ DFE
Or ∠ DFE = ∠ ABC …………eq(1)
⇒ Similarly ∠ FEB = ∠ ACB …..eq(2)
⇒ In Δ DEF and Δ ABC from eq(1) and eq(2) we have
⇒ Δ DEF ∼ Δ CAB
⇒
⇒ ar(Δ DEF) : ar(Δ ABC) = 1:4
Hence proved
Question 4.
In Δ ABC, XY || AC and XY divides the triangle into two parts of equal area. Find the ratio of
Answer:
Given, XY || AC
Need to find the ratio of AX:XB
⇒ ∠1 = ∠3 and ∠2 = ∠4 [corresponding angles]
⇒ Δ BXY ∼ Δ BAC
⇒ ……….eq(1)
Also, we are given that
⇒ ar(Δ BXY) = ar(Δ BAC)
⇒
From (1) and (2)
⇒
⇒
⇒ Now, -1 =
⇒ =
∴ AX:XB = √2 -1 : 1
Question 5.
Prove that the ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding medians.
Answer:
Need to prove that the ratio of area of two similar triangles is equal to the square of the ratio of their corresponding medians
⇒ In case of two similar triangles ABC and PQR we have
⇒
=
⇒ Let us assume AD and PM are the medians of these two triangles
Then
⇒ =
=
Hence, ar(Δ ABC) : ar(Δ PQR) = AD2 : PM2
Question 6.
Δ ABC ∼ Δ DEF. BC = 3 cm EF = 4 cm and area of ΔABC = 54cm2. Determine the area of ΔDEF.
Answer:
Given, BC = 3 cm EF = 4 cm
Also, area of Δ ABC = 54cm2
Need to find area of Δ DEF
⇒ since, Δ ABC ∼ Δ DEF
⇒
⇒
⇒
⇒ = ar(Δ DEF)
⇒ ar(Δ DEF) = 96 cm2
Hence, the area of Δ DEF is 96cm2
Question 7.
ABC is a triangle and PQ is a straight line meeting AB and P and AC in Q. If AP = 1 cm. and BP = 3 cm, AQ = 1.5 cm., CQ = 4.5 cm.
Prove that
(area of ΔAPQ)
Answer:
Given, AP = 1cm, BP = 3cm and AQ = 1.5cm,CQ = 4.5cm
Need to prove (area of Δ APQ) = area of Δ ABC
⇒ it is evident that Δ ABC ∼ Δ APQ we know that
⇒ =
= =
⇒ Area of Δ APQ = Area of Δ ABC
Hence proved
Question 8.
The areas of two similar triangles are 81cm2 and 49cm2 respectively. If the attitude of the bigger triangle is 4.5 cm. Find the corresponding attitude of the smaller triangle.
Answer:
Given, area of two similar triangles as 81cm2 and 49cm2
Altitude of the bigger triangle is 4.5cm
Need to find out the corresponding altitude of the smaller triangle
⇒ Δ ABC = Δ DEF
⇒ AP and DQ are corresponding altitude of triangle
⇒ =
⇒ =
⇒ =
⇒ =
⇒ =
⇒ = DQ
⇒ DQ = 3.5cm
Hence, the altitude of similar triangle is 3.5cm
Exercise 8.4
Question 1.Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.
Answer:Need to prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals
![](data:image/png;base64,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)
ABCD is a rhombus in which diagonals AC and BD intersect at point O.
We need to prove AB2 + BC2 + CD2 + DA2 = AC2 + DB2
⇒ In Δ AOB; AB2 = AO2 + BO2
⇒ In Δ BOC; BC2 = CO2 + BO2
⇒ In Δ COD; CD2 = DO2 + CO2
⇒ In Δ AOD; AD2 = DO2 + AO2
⇒ Adding the above 4 equations we get
⇒ AB2 + BC2 + CD2 + DA2 = AO2 + BO2 + CO2 + BO2 + DO2 + CO2 + DO2 + AO2
⇒ = 2(AO2 + BO2 + CO2 + DO2)
Since, AO2 = CO2 and BO2 = DO2
= 2(2 AO2 + 2 BO2)
= 4(AO2 + BO2) ……eq(1)
Now, let us take the sum of squares of diagonals
⇒ AC2 + DB2 = (AO + CO)2 + (DO+ BO)2
= (2AO)2 + (2DO)2
= 4 AO2 + 4 BO2 ……eq(2)
From eq(1) and eq(2) we get
⇒ AB2 + BC2 + CD2 + DA2 = AC2 + DB2
Hence, proved
Question 2.ABC is a right triangle right angled at B. Let D and E be any points on AB and BC respectively.
Prove that AE2 + CD2 = AC2 + DE2 .
![](data:image/png;base64,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)
Answer:Given, ABC as a right angled triangle
Need to prove that AE2 + CD2 = AC2 + DE2
⇒ In right angled triangle ABC and DBC, we have
⇒ AE2 = AB2 + BE2 ………..eq(1)
⇒ DC2 = DB2 + BC2 ……….eq(2)
⇒ Adding equation 1 and 2 we have
⇒ AE2 + DC2 = AB2 + BE2 + DB2 + BC2
= (AB2 + BC2) + (BE2 + DB2)
⇒ Since AB2 + BC2 = AC2 in right angled triangle ABC
∴ AC2 + DE2
Hence proved
Question 3.Prove that three times the square of any side of an equilateral triangle is equal to four times the square of the altitude.
Answer:![](data:image/png;base64,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)
Given, an equilateral triangle ABC, in which AD perpendicular BC
Need to prove that 3 AB2 = 4AD2
⇒ Let AB = BC = CA = a
⇒ In Δ ABD and Δ ACD
⇒ AB = AC, AD = AD and ∠ ADB = ∠ ADC
∴ Δ ABD ≅ Δ ACD
∴ BD = CD = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAgAAAAdCAMAAACgyt2RAAAAAXNSR0IArs4c6QAAAGZQTFRFAAAAAAAAAAA6ADo6ADpmAGa2OgAAOgA6OgBmOjpmOmaQOma2OpDbZgAAZgA6ZjoAZrb/kDoAkGY6kJC2kLbbkNv/tmYAttv/tv/btv//25A627Zm29u2/7Zm/9uQ/9u2//+2///bzXp2qAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAa0lEQVQoU42PyQ6AMAhEB/d932u1/f+fFIyJJnrwXRjCEAbgRgXkxjOgnQ6LD9i6AoYM2MIepuhPsTfR1MI2lKy5Nz62f0i6+GH9tKwlUcoTU3RQDucRJNzJwkkFJRbpr6q5av7E5HKexYsDsfcEkAyU8qwAAAAASUVORK5CYII=)
⇒ Now, in Δ ABD, ∠ D = 90°
∴ AB2 = BD2 + AD2
⇒ AB2 =
+ AD2
=
+ AD2
3AB2 = 4 AD2
Question 4.PQR is a triangle right angled at P and M is a point on QR such that PM ⊥ QR. Show that PM2 = QM.MR.
Answer:
![](data:image/png;base64,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)
⇒ Let ∠MPR = x
⇒ In Δ MPR, ∠MRP = 180-90-x
⇒ ∠MRP = 90-x
Similarly in Δ MPQ,
∠MPQ = 90-∠MPR = 90-x
⇒ ∠MQP = 180-90-(90-x)
⇒ ∠MQP = x
In Δ QMP and Δ PMR
⇒ ∠MPQ = ∠MRP
⇒ ∠PMQ = ∠RMP
⇒ ∠MQP = ∠MPR
⇒ Δ QMP ∼ Δ PMR
⇒
= ![](data:image/png;base64,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)
⇒ PM2 = MR × QM
Hence proved
Question 5.ABD is a triangle right angled at A and AC ⊥ BD
Show that
i. AB2 = BC . BD
ii. AC2 = BC . DC
iii. AD2 = BD . CD
![](data:image/png;base64,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)
Answer:Given, ABCD is a right angled triangle and AC is perpendicular to BD
(i) consider two triangles ACB and DAB
⇒ We have ∠ ABC = ∠ DBC
⇒ ∠ ACB = ∠ DAB
⇒ ∠ CAB = ∠ ADB
∴ they are similar and corresponding sides must be proportional
i.e, ∠ ADC = ∠ ADB
⇒ ![](data:image/png;base64,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)
∴ AB2 = BC × CD
(ii) ∠ BDA = ∠ BDC = 90°
⇒ ∠ 3 = ∠ 2 = 90° ∠ 1
⇒ ∠ 2 + ∠ 4 = 90° ∠ 2
⇒ From AAA criterion of similarity we have in two triangles if the angles are equal, then sides opposite to the equal angles are in the same ratio (or proportional) and hence the triangles are similar
Their corresponding sides must be proportional
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒
= BC × DC
(iii) In two triangles ADB and ABC we have
∠ ADC = ∠ ADB
⇒ ∠ DCA = ∠ DAB
⇒ ∠ DAC = ∠ DBA
⇒ ∠ DCA = ∠ DAB
⇒ Triangle ADB and ABC are similar and so their corresponding sides must be proportion.
⇒
=
= ![](data:image/png;base64,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)
⇒
= ![](data:image/png;base64,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)
⇒ AD2 = DB × DC
Hence proved
Question 6.ABC is an isosceles triangle right angled at C. Prove that AB2 = 2AC2.
Answer:Since the triangle is right angled at C
∴ the side AB is hypotenuse.
⇒ Let the base of the triangle be AC and the altitude be BC.
⇒ Applying the Pythagorean theorem
⇒ HYP2 = Base2 + Alt2
⇒ AB2 = AC2 + BC2
Since the triangle is isosceles triangle two of the sides shall be equal
∴ AC = BC
Thus AB2 = AC2 + BC2
AB2 = 2AC2
Hence, proved
Question 7.‘O’ is any point in the interior of a triangle ABC.
OD ⊥ BC, OE ⊥ AC and OF ⊥ AB, show that
i. OA2 + OB2 + OC2 – OD2 – OE2 – OF2 = AF2 + BD2 + CE2
ii. AF2 + BD2 + CE2 = AE2 + CD2 + BF2.
![](data:image/png;base64,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)
Answer:Given, Δ ABC, OD ⊥ BC, OE ⊥ AC and OF ⊥ AB,
Need to prove OA2 + OB2 + OC2-OD2-OE2-OF2 = AF2 + BD2 + CE2
⇒ Join point O to A,B and C
(i) ∠AFO = 90°
AO2 = AF2 + OF2
⇒ AF2 = AO2 - OF2 …….eq(1)
Similarly BD2 = BO2-OD2 …..eq(2)
⇒ CE2 = CO2-OE2 …..eq(3)
Adding eq(1), (2) and (3) we get
⇒ AF2 + BD2 + CE2 = OA2 + OB2 + OC2-OD2-OE2-OF2
(ii) AF2 + BD2 + CE2 = (AO2-OE2) + ( BO2-OF2) + ( CO2-OD2)
= AE2 + CD2 + BF2
Hence, proved
Question 8.A wire attached to vertically pole of height 18m is24m long and has a stake attached to other end. How far from the base of the pole should the stake be driven so that the wire will be taut?
Answer:Given, height of a pole is 18 and wire attached is 24m
Need to find the distance from the base to keep wire taut
⇒ Let AB be a wire and pole be BC
⇒ to keep the wire taut let it be fixed at A
⇒ AB2 = AC2 + BC2
⇒ 242 = AC2 + 182
⇒ AC2 = 242 + 182
⇒ AC2 = 576-324
⇒ = 252
⇒ AC = √ 252
= √(36 × 7)
= 6√ 7
Hence, the stake may be placed at a distance of 6√ 7m the base of pole
Question 9.Two poles of heights 6m and 11m stand on a plane ground. If the distance between the feet of the pole is 12m find the distance between their tops.
Answer:![](data:image/png;base64,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)
Given, BC = 6m, AD = 11m, BC = ED
And AE = AD-ED = 11-6 = 5m
BE = CD = 12m
Need to find AB
⇒ Now, In Δ ABE, ∠E = 90°
⇒ AB2 = AE2 + BE2
⇒ AB2 = 52 + 122 = 169
⇒ AB2 = 169
⇒ AB = 13m
The distance between their tops is 13m
Question 10.In an equilateral triangle ABC, D is on a side BC such that
Prove that 9AD2 = 7AB2.
Answer:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAWAAAAD8CAIAAADG0OmXAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsMB2mqY3AAAVFNJREFUeF7tfQdgXMd1Lbb33rAAe++9F1ESJYoi1Xu1HceJy7cTy/6Ok/zkR3aSn2I7lh07lmR1ibI61QvFKvbeG9iJtr1ge/9n3ttdLECQWCy2AvMMU4vFe1PuzJw3986993BSqVQdvagEqASoBHqSAJeKhUqASoBK4GoSoABB5waVAJXAVSVAAYJODioBKgEKEINjDiSpQWlwDHTZekl3EGUTdckrcvpiv/388ls7rSWviVYwaCRAAWKADDUOo3ae8f7Lexd+9fHFy47QAOkV7UalJUABotIjUKT6A5H4hmPOxeNUapng04POIpVKixnsEqAAMUBmwHlbaP8F34OL6ueNUa074gxGEgOkY7QbFZUABYiKir9IlcM2ueWEW8jj3DJNt2yS5rIzvL3JU6SyaTGDWgIUIAbC8Hv88c0n3Msna3QKwcIxqtEm6cf77QOhY7QPlZYABYhKj0Ax6j9yueNkazAcS3283wH9glNXt/ec75yVmiqLIdzBXQYFiJof/0Qy9dlhZziW2HXO+/SG5hc2t7W4wraO6KcHHTXfN9qBSkuAQ4O1Kj0E/a3/siP8+O+OLhqnWTlTx5YViSVf2NQWiiZe+/5UhYTX3wro84NYAnQHUfODD+tDIJp8dKl52UQN+7Nimu7uuYaz1uDWU9RUWfPjW9kOUICorPz7W7s/nPhwvx1WyREGcW5ZM0YoFGL+2r3WZB31vu6vkAfz8xQganv0o/HUsonqb9/UKBN1USWG6SU/um34nFHKUCRZ2z2kra+oBKgNoqLi73fl8UQqGk+KhTwuji66XvFkCsYIkYDLv/Jv/a6XFjBIJEABYpAMNO0mlUAhEqAqRiFSo89QCQwSCdAdRE0NdCoVCPhsNntdKmU0GWUyeR2HQnxNjWCtNZYCRM2MmMftevmVV155/W2LK1zHqavXSB65/64/+7NvaHX6mukDbWitSYACRG2MmMXS9pOf/O17u9tTo1ZzVUPq6jgpXyv3wqerpmme+vUvzA34hl5UAsWXAN2gFl+mRS8xkUj85te/fmu3kzvjW/KR82XaBvIzbDZnxnc+OtTx69/+LhwKFr1SWiCVACRAAaIGpsGxIwc37G1KDLlOahxdl4ylEviJ1iUTYpWRN/Hetz7a1Nx8uQa6QZtYgxKgAFEDg9Z06qQtIhPrhtclYjBPZlqMD0m+eogzIrx48VINdIM2sQYlQAGiBgYtHo+l4O7E49d185sGRHA4dTxhLE7zR9XAONZiEylA1MCoDR8xWs0PRzocdTxBl+ZyuKmwT84JDG0010A3aBNrUAIUIGpg0KbPnDVnlDzZuiMRDnAEkjouHz8cvjiZiETPfnHd7PEjR46ogW7QJtagBChA1MCgyeSKRx55bIywxb3/9ajzYjIawk/Ift518B2u7cD4RfdwxOoa6AZtYg1KgPfkk0/WYLMHXZPbk6YDLQnHsS9S1kNJb0vSfkzu3D2M366Z+YBHOU3M505slPYQsDXo5EQ7XGQJUEepIgu0FMW1uSO/+OjSzgsRvXNr3flP3bZWnkC4fMVti1c9/MUZ3vZTTqWY+52bh9w5xwiTJb2oBIooAQoQRRRmSYqKxJPPb2z703aLTMSdUHdQGb4UjkREYsmtt98zafKUC9bgM182H7zkq1cJf7By2E1TtSVpBC10sEqA2iCqfeR3n/GuP+qMxFOzh/KGyGORaCSV4sTj8YDfH4nERhjFjy01T2iUWTzRP25s2UnpMKp9PGusfRQgqnrA2j2RD/bZz9uDM0apZpoTghTCtLgCgYDD4cL/OplKJZJ1k4fKHltSP9wgvmgPP7Oh9fAlX1V3iTaupiRAAaJ6hwvKxccHHLvPeg0K0apZBiXX5/b6pFK5SCxCoxOJODwp4VcJJ6lZI5UPLa43KQWnWgPPrm85Z6GhGdU7rLXVMgoQ1Tte+893fH7Igez1N0zWzh4u87icwWBQbzRJJDI0GlpGkkk3CZAA9d7S8eoHFtYrpfyDF31/3NDS4oxUb8doy2pHAhQgqnSsYFN4f6/tvDU0bZhixXQ9Lx50ujwABVN9g0QuTyaTUDGynCbJZIrD4dw0TXffAqNQwN162vvi5ja7L1qlfaPNqh0JUICoxrFCHtovDjt2nPYYVMJbpuvGN8gsVpvf3yEQCPUGg0QswaYBAMEEbqUPNmGQEPA4q2fo75pjQJdg13x1a3tHmMZoVOP41lCbKEBU42BBTfhovx0nF0vGqxeNV3O4HIfD7uvwqXCpNXw+oraIDSKF2K0cxwdkuJYIeXfMMa6aqSf2i/2ON3dY8KEae0jbVCMSoABRdQNl8UTW7rGBenfKEDm2Dyop4rQSbrczFAzqDEa5XAltAhuHBPAAB55dm4/v1DLBvfNNN07W+kOJd3ZZURQsFPSiEihMAhQgCpNbqZ4CEc76o65tpzwGpXD5VN04syye4PihXXi98XjUYKyXIlEts21gVIzuzcAX8UTSpBI+sNC0aJzK6Y+t2d7+yUF7qZpLyx3oEqAAUV0jfOSy7/29RLlYME513UQNOcbkcHF+4evw8vkCrVYvkUrI/oEBiB6Jl5mDzyTcIh5eXD9zuKLdHX15S9vG467q6idtTY1IgAJEFQ2U1RN9b7f1gj00sVG2crpeJeVBZeDxOC6Xo6PDK1colSq1gA8nKQ7yxCTijA2ipws6BR4c1yB9ZEn9hAbZJXv4xU1tu854qqirtCk1IgEKENUyUFjSm0+4vjrt0ckFy6doJjRKYV+EuQHbAbfbFfT7dXqDTC7HBoELzABAwEiZc4rRrRsEI5Kp6cOVDy2qH6oXn2oPvLC5DduTauktbUeNSIACRLUM1PFm/7u7bbF4au4Y5fWTtVj7RL3gcGCb7PC4ozEYIEwwQMC3mp8GCOaY8+rhm3CjgoayYKwKGGFUCg9f8r+0uY06WVbLeNdIOyhAVMVAOXyxd/fYLjrC48zSW2foVRI+NhRoGZfL87rdHV4Pj8sDQY5YIobXJJJTkh0EVIzejifgHIETjxsmae5fUK8U83Y0eV7c0tbqok6WVTHoNdEIChCVHyZYGzcdc2054VbLeDienNQog6MU2ywuj+N2O7xeD5QLGCDgAQG1gsfsIOKMitFr/gcADZfLuWW69p75JrGAt/GY67WtbTjdqHy3aQtqQQIUICo/SidaA2/tskYTyVkjFDdO0UIvyO4M4EPtdrn8Ph8MEDBSsn9gASIJZeMqRspuXQJGiPjc22bqV88ycLlcBIAhu4QvjFgvelEJ9CIBChAVniIuf+ztXZZmZ3ikQXLbLINCwiMLn90+cLmgzPLCABGJ6PRGuUKBEAxABDFScrkJIETeLlCxREom5t0zz7himgano+/str6z20adLCs89rVQPQWISo4SdgSbjru/OukBLtwwRTuxUU7OLjMXAALWhw6vGzqCVqeVSmUpJn6Th5TWdRyEa+GnNytEZ++AEVoZ/7759UvHa8LRxJpt7fDmhpGikv2ndVe9BChAVHKIkL6BUS5SM0Yob5qqg26Ru14BEDjg9LjdUplCpdayBghoFXw+dhAACERsJfNTMtJ9REVmjejBRfVzR6t9ocRLm9u/POIkvlj0ohK4igQoQFRsaniDiTd3WS87QkO0otuRDwbKxRVLFegAFymNVqtQqtBQBkBwisHaIEhGqb62HubPkUbJw4vqpw6V2TuQpa4Vbt19LYTeP3gkQAGiYmO96YTzqxNuxF8i5mLiEFm33T78JcPhMAwQkUhYy1goYYBg24qtBBcAQVSMfO2UuZ0ERkwaIn1okXm0SQLbx7MbWg9coA5UFZsGVV4xBYjKDNDptuBbO63heHLGcPkKpKJm3KJyL2wTfB0er9uFg0ytVieTkSQx7A3wiSA7COgYyQLTPSAkbPZoOFmahujEZyzBZza0nGz1V0YQtNbqlgAFiAqMTyCceGunBSESZrXwzjlQLhjjQtcLaoTXAw3DKZHKkANCKBRm78m4WsNnqs8qBlsJHovFk0smwIHKqFcIDl7oeG5D62VHuAKyoFVWtwQoQFRgfDYcd3110s3nc26epp00RHG1qEx4WMNFSq3RqFQasqozcMD4QXAzKkaB7SdBn8nUzVN198w1IuXE9ibv85taYZUosDj62ACVAAWIcg/smfbg2zstgUhixjA51iez8ru3AecX0UjU7XFFgkF4WCtVKlgksy9/1lEK+gV76omAjcL6wIR7cOA9dftso0TIXXfE+fKW9o4QdaAqTJwD8ykKEGUd11As+eZOywV72KgS3jUPr25BjweVAAgkifG4XIAOtUYH8l7sF9INTaU9KYEOxAxxrXCt3ruG54E298w1IHUVqD2RJnfNNks4RrPU9S66QXIHBYiyDvSGI66tpzx45d8yTTd5iII5teyhAQAIL+yTLqdYIoWKIRSJcowU6VgMLGKSVKquv4sZICMR8RDNdf0k4mQJ4wisp2yoGL2oBChAlG8OXLCF3tlj7Qglpg9TwC0KaV+uYWX0eT0441Sp1TBAdFMhWCMl2s0klSpC+wEHGjkfGagWjFWDhuN1OFkeoFnqiiDYAVAEBYgyDSICH97YYUE6Bp2Cf/c8o0YuuFokBRZ/LBbDCUYw4NdodAji7HYnD+ZNoEsdePeKZi9AHooGjQgUfjOGKz3BODJHwMmyTKKh1VSxBChAlGlwNhx14uQC3lC3TtdPGSq/im5BGoMcECDmdbucUCvUGq1SCYDo4u/A58JRigvdBImts84R/e8GnCNGmaSgAh5rloK2B4cayB/R/2JpCTUtAQoQ5Ri+i/bQe3vs7kB8xggF/CYRS3GNQEwYIJCi1um0i0RiGCBEElEuCjDRnFxksiUqBmg5i3rFEsnJQ+WPLzUjSx0sqX+kVMBFFW8tFkYBouSjhjBKmP1Otwe0csHdc40GpeBaMZRQMDh1iL+AiZJJUqtlzQ2dF3JSkou1QTBpKQs95ryy50xG7NScUarHrzMbFAK4V4Lms6k9UHIZ0QqqVQIUIEo+Mkji9NUpNzbwt85klIsrvKpzW0BSRcXjME8G/T61VqfSaK4I6UaGKHIhGyULEAV6QVyl30yMaGrRODXowuUS/oGLvuc3tbXRLHUlnyZVWgEFiNIOzAVbeO1em80bmzlSiXRyIgESvVzr4AFRWMFgwO1ywO6gVsOHUs1kjupywY2SAATIta74U1E6g6MRgM7yKdoHF9SD73P7KQ/sEU4fzVJXFOnWWCEUIEo4YDiFXLvHinTVyNQCTt16tRDezdeuD0eYSDDndDgEQjH0C4lEmkx1NzQQBYMAREbFKEEPgGLIUrdyhv6euSbseEAF/PJXbQghKUFVtMiqlgAFiBIOz4Zjrs0n3JFYEomqpw6VXxmy2WPdfl+Hy+lQKJTYQXQ3QDAPcBDOiYDONPteMRwhemoHTCdSMe/2OYZVswzQjz45YH9tWzu+LKG8aNHVJwEKEKUaE5xcfLjf3u6JzBqlRDo5qZDb6/aBSWafgAECGAH7pFqr7fEUE/sHsoOADYI4PJZwxUKFUUv5984zIpWuP5wEDzCiSEolL1puVUqAAkRJhgV2vg/32Q5d8qll/Ntn64doxbHelAu0g81SC6I9uEgixBsqRs8AQfCBDBzMmYwnZXHNlJ0CYSM+QQV8/0LT/LEqVyD2xk7rB3upk2VJ5kx1FkoBoiTjglS0G497oFyAYhOO1eTgIo83PZZ9IOBz2m0CgQA7CJKltqfHONAxMjaIa5Nr9b9vqB9qxXA9oQKeOkxhcUegaKw/RqmA+y/a2iiBAkTxx+myM/zhfhuyuZGTiylamThNk9VrTVAx4EPpdNilcjnyUPL43B4BgrFRsqcYhL+3VPuHTHNZjAAJ8GNLzGPqpZfsoVe2tO864+21O/SGASABChBFHkRoEh/vt+8716GV82+bqRthkMAfOp86GD5eYoCAlxTcq5EG4mpJ7ZlTTuIpVaxgrV6bxzhQJacPJ06WjTrRqTb/i1taj1yiWep6lVzN30ABoshDCAa99UdcSDAP5WLWSCXxO8pDuSCGBA4nEg4jBAOWBeJCSQwQVzlWZFJC4AES7p2P6lKMLqIX8I+YO0b1+NIGvUJ48KLvxc2t5yyhYpRNy6heCVCAKObYtDDKBc4vEHNx/WSNIsPBm08d2BTARcpht/H5AsCDXA6fy6tCCy/L31vKU4xuzWbzVy2ZoIY9Ank0tzd5XtrSimOafHpH76lRCVCAKNrA4czxk4OOnU0enVKwaqZ+lFHaJ247KA6BgN9ht0qRJEar4wt6yGSbNQuwWedggyjbDoKtGg5USDy1fKr2vnkmqZC3/qgLgeEIQiuaEGlBVSYBChBFG5CdTd7PDzlhcICT8pxRRLnIf/tPckwmEgjixCVXKrVa/bV5NzME36zfUqnNlF1ExFIBr56pv322ATD1MRyotrYFo9TJsmgTqaoKogBRnOFod0fg+HDeFsRZ4E1TdEppvicXbPUAiGg0CgNELBaFDyWxUGZYMLq1j9U6MpnvmYTYZcUHUjswAlTAd8413jwNdIF17+62vrnDijPd4oiSllJNEqAAUYTRiCaSnx1yIGTTqBTeOl2HhCt9XS3EABEK2q1WGBdgoJQrOmlyemwfyLUIfy+CPa6CI0Xo1TWLwMGnTi64b4Fp6QRNOJZClrr399mpI3apxV7+8ilAFEHm+893gCkbL1AYJheMV+MFm79ykd1BhANBh80qFiNJjFYg6KTJ6d4+ZgsBHGFPMYgmkt8pSRH62bUIUPg1akSICp8zWukNJV7d2rYeVMB5ntkUvTW0wNJIgAJEf+UKspn399jP20KTh8hXTNPBvN9rzMWVVQJRfGDa87jkcqVOb8DZ6NWbRfAAaekYagwYOiqp/GOjNApUwIvrJw+R2bzRF7e0bT/t6a9A6fPVJAEKEP0aDbxFPzvk3HzChYAFUEuMb5CF+26ug34B04PLbQdPr0yhAFUvVIdrN4s1UrLMGD1nzu9Xt/rwMEg0kAUHmSxHm6TnrUHQhWM/1Yfn6a3VLQEKEP0anyOXfMgHAzvhkvGapeM1CIsuoDgkgAmFQjarFaZHWCiZNNbXMvihDj7SWrIE39faaxTQlkIewT5izijFI0vMQ/TiE60BYMTpNpqlrhBJVuEzFCAKHxSnL/r2LitOLiY0SFfO1CukvGtni7paTVjq4VDIbrOKhCKEYICnt5dNAdJSkh0EvBLIVXgHivQkDC6ROMlS9/AiM3ZS2EEg2y2lAi6SdCtcDAWIAgcAFvsvDru2nPTg5AKmh/FmaTha8FpNBfw+j8uJGC2dzpCPb3ZWxeg3sVaB3e/2GDAC5hDk1Lt3vgkR7jjQeW4jpQIujmwrWwoFiALlf6LF/85uK3wQ5o9RXT9ZW/AJH5OlNgaWPQRiyGRyncGYx6aAsO/hFAMaBmuEqIYLDcGmBqwfd8w2yES8L486X9zc5gtV0oZaDWKp9TZQgChkBOFc/MYOK6j0Rhult80yKCQFKheoGx7W4UjYbrNgwcuRZk6jveYRRrq1XD4fpk3YIMgpRpUgBOOILeRz7phjAPMojCQf7Xe8urUdfMWFiJg+Ux0SoADR53FglAtCk6VXClZM005sxMlF4WsAuR0iobDN2i4QCoEOoi48vVdtG58J1sL2gTnvwKu77N6UV2kajniVYv69oAKeqIXS8e4eK7LUUSrgPk+yqnmAAkSfh+J0W/DtXRbM/jlMPpiClYtMxSlwcCJLLWK0dHpjnrsBkrSWAYUkidfqcxdK+gBC3fUKwUOL6ueOViFGA1nqEOHaV8+xkraQFp6/BChA5C8rcqc3FP/T9vaLjvAwvfiuuUa4RRV2csHWCjUB2R9cbidOMaRSqcGYjwGCeZA55sTeIZ4kSaX61ofS302cLLXCx5fWI5e3yxd7basFJonSV0trKL4EKED0QabYz6877Nx6yqOVCm6Zph8Ht6j+KdhMjFbEYbUisEImlxMDRH7HlnyS9p5Dzg4K8OvuQ48Lv5WlAn78uoax9VIQcyEqfHsTzVJXuDwr9SQFiD5IvskSfGunBToF8sHcNE2LM4Q+PNzTrQQgIhGrtZ3H56s0WrFYmmcUB+5nVYw4VIyqvLCrIVTAjfKvXUey1J2zhUDPdeQyzVJXlaN19UZRgMh3wHBit2Zb+2VHBBFKd88zKMRQLvJ99mr3AQ6CgSCy1IolEr3BmL+tkThKpfl7+x4Z1t9W5/s8NjjQv5C592vXNcBb5GQLoQI+Zw3m+zy9rwokQAEir0HAXmHdEceOJi9ONFfOgFuULBbvLzywxxAejzMUCIjFEoPRfNUklF3biJczyWvNYdn3qtrRAKZcNHLBWNUjTJY6OFk+u761zU2z1OU166rhJgoQeY3CWQtOLmzhWAL5YJAlBdjQf8MgAAIxWvCAwD4CLlJaDZKv5FcqYj8y3DkJwp2Dp6rlmPNKaTJZ6uqWTdLcv8AkEnKRyRL7CLefUgHnNfEqfhMFiN6HIBghysUlR9ikFt0z3ygX84vivkgAIhq1WtrhFokstRKZtPempO9I5fD3MjuI6sUH0jroYshSB2CFkyX2PRtPuF7Y3Aqp5t1femPFJEABohfR4+0Mt6gdZzwiAWclArrN0v7bJrNVIojTabcLRSK9wUTyO+S5g2DOR7PkWniquvGBdBcOVAoxDwCB0x8cBiFG/pWv2vuvplVs3QyaiilA9DLUZ9uDiLkIRBLThipunqrjkmVchNnBGiBAk4MwLWSRMpjq80cHVM9oGAx3Thzv4WI0qAh96qUIHMhq5AJQASNLXSCS/GCf/c2d8DcrfcW0hn5IgALEtYQXiiXWbLdAuYAR/r6FRpWkOMoF0QlIjFYcHtbABYlUpkUQZ1+WChfJ51kjJRyl+vJgP6ZKfx9FM2EwMamFDy5CljoFqIDf3m37aL+NYkR/JVvK5ylAXEu6cIvadcbD53JunaEfZ5YVMf8jG8Rps7TD3AiiPdDk9GkjAHRIqxi1s4OAoAlGgApYJ350ScOUoTKLJwr83XjMVSsYV8qVWKVlU4C46sDgxP69PTZvMD51mBxUF4AJ9tCuWFc4HHY47MhP2ycPCLb2rA0iDoW+mI0qVueuWg6h+UymxtZLvnZdI7LUISL2la/adp+lTpYll3xhFVCA6FluSKP2p+0WTF/wUN4334gkKEVEB9Ye6fN6wZQDC6Wx3kzern1Z6IwbBBeHF8l4ucm1CptnuU9hI4ZjIOwgvrbUPFQrOdkaeGlL2/Fm6mTZf9EWvwQKED3LdN0RcnKBNbtqhm5cA/b/xbFNZiuD/wIMEPCMgouUzmjsCzgwOwgOBwGdaBUh+CZfVP85Rhc5MxhRN3uU8rGl9Tg83n/e9/zGVsBx8Sc4LbF/EqAA0YP8kMN+7R6boyM2bbj8xik6JEHpT8jmlRUQD4h43Gppw58USpVcpuyTAYItEOEY+BeGDELPWWP4QNqPHRmiSRaNVz24yKSVgwrYiyx1FkoF3L/1XPSnKUB0FykCNN/YYTlrDRqUwnvmmXQKQXHRgbzumRgt8GiByNtgNJH8cX2/cug5a8APosf+QbACHvfGSdq75xpkYu6G4+4XCBUwdbLs+2wo2RMUILqLdsMxFxi6YWzHycWERhn+3Nf9f6+DBXOD39fh83kBECaTubD3fzpvLWIxaspI2U04kLNUxLt1pn7VTIOQx/nsoAMOVP4wdbLsdRKV6QYKEF0Efcke+mCfrd0dRUA3Ti4kAm4ptg8wHNhsFvhBiMQiGCAYDOrzKkfOGDzI2CD6/GyZJld+1SC7jFIiuGuOEQJHZ97fa39rpzXUjyx++VVL78pLAhQgOsUEUizwXJxqCRjVQmSLgnNUvyM2exgD6BdY1VZLK1ABNDkKhaqwBc5mnSP5IGobH4iIgBFI8Akq4MXjSZa6t3ZZPjpgL4yFKK9ZT2/KWwIUIDpFtfmEG4QOmKxQLkC0WdiLPR/Jw7IIAwR8GRCCIRAICrBQohY2by0yUOWRJj+fRlX4HpwrD9GKQPM5d5TS6Yut2dq+/qiz6Nu3CneyBqunAJEeNCgXILBvdUZmjVKCAEYq4hbAwZvnBABPb4fXg2MIU31Dno9ceRtyxuDLKmHfK7gXuQ/CPAwagceWmOGZ1uIKv/RVGxJwFKVkWkjBEqAAQUQH7ob39tqOXfYbVMLbZxvMamGsNGYyNkbLYbPGYjGhUIQjjIL3KXweicdIEHbO0rS14DnVjweBETAMwxEb+T4RJvf8xpaDFygVcD8E2u9HKUAQEW475d50nCgXYOieNlxBqCb6bjXMcyzwzre2twEm5HLC05t/mrlu5fNgpIT3NxAiOQCsEJ2dQ9b8WSMVjy42jzCIkcPy2Q2tp9tplro8J1fxb6MAUdfsDCP0uNkRhnKxfLIWtHGlI3ohZsUY8aHEB53BALKcgpEI5FrwqExUB8F3EScmBAL5LxyrArNGg0a09xyy1DVjjIpYBS0qfwkMdoAAKdaH++wHLvgQhrx6lmGITlzSLCaY/YGA3+vxwEJpamgsbPvAnlrwuXwMHoMPA2sLwThZoo/LQAU8z6RVCLae8j67ocXeEc1/WtM7iyWBwQ4QO894EXYRjSVvmqKbNYIoF6VLTwDPYsx8GCAi0YhQKDQgixSXW8gOglF/iKMUVAyEPZWuxcWaZX0vB7iH8NkVM3S3z9JjT7fhqPuFTW2IrO17SfSJfklgUAMENq4f7rMhHwxRLqZqZeISKhfMKBFpIwklDAdSqUyj0fQnhILEYpCoULxui5JDt1/TqBQPQ9GAo9odc4wgQBUwTpavgQo4OnAssqUQWtHLHLwAgYP3Tw869pzrqFeLbp2hG2GQwEhZdPnmFogdBLwn7XYLNhI6g1EoEvfHxYk4StUxO4iS2VNLKo18CgdHkVLCIw5UE9SxZBLn0PBkK6kOmE+rBtU9gxcg9pzt+PSQI5JI3jhZM2+0iqyz/qzX/GZNKBT0uFx485vqG0lCh35UydJzMgTfbGLr/mxH8mt9Je6CPyWogEGrASrgQCj+5k7rR/sd5RiqSnS2CuscpABx2RFeu9d62RGaOVyxfKpOXnLlgkRwYlo77dZIOMwXCAwmUz/z38IPAlFeKDORIPy9AxMemBUDjGjUikDzOWWY3OmLvrqtff0xSgVcJjAZjAARiSdhmNx51mtSiZCoepRRgm/KIG8sZmt7O6InJBKpVqtjss4WvmnhMa7WaDby1vajmDL0uwhVxOKpkQbJ15c1jKmXgZjrhU3tu89QJ8siCLbXIgYjQIAADrFAsIEh/zoSlpRHucBI4DwSPFrQCHQ6g0gs6Q86oDQ4SsEGgS1EvHRuG71On3LdQLLdJpPjG2TfuL5hqE50wRZ8ZkMLzVJXBvEPOoBodUfW7rXBLWraMMWKaeVQLtKjyOGEwyEXDBB1dSZzA/wg+qNIY8Gw+SDwgVBj9MOWUYZJVpQq0EVIbPow0IU3IJfPiRb/79c1n7PSLHVFke5VCxlcAIFzivVHnNtPe40q4U1TtGPqpeWLKYYBwuEIh4I8uEjVmwvLIpU7jFycYsDMSXylBsvJH3H44NTBovzIYrNaKjh0sePpL5tplrqSIsTgAoiDF31r99lwMrh4nBr6BXNGWFLxZncPxFhgtbaDrVcslWp0ugJdpHIam5P5nuXvHRQXvEYRg7Jskva++SaJkA8/tz982UKz1JVu7AcRQLR7ou/ssTU7I5OGyG6ZoUcSxLKYJtNjBzCyW9uRzFqj08NI2d8RJQTfSH1PcCdJuHMG0QUqYKGAc8t0/e2z9fC23HLCjYAucCMOIhGUsauDBSBgyNtw1Ln9lMcgF8Bpcmw5lQtmOCORiMvpAEzU1zfgAKL/VgOG4JukhIgnwZ0zWHYQ7NJAnj2pmHvnHOONU7RI2/HlEVABt1EHqlLgxmABiEMXfe/usWIhzR+jWjZRk2aTKIVEeyoT1kSPyxEMBpDlhTVA9H9Jw5bB0Pcy3DnkGsCeED3IFIivkvIfWFgPbTEYSX603/4mnCxZkhB6FU8CgwIgbB3Rd3ZZW53RcWbprbP0UiGvvHx1ZOnarJZYNCoSibQ6PeyL/QYIkEqwKgaCowm51gB1pLzWTAccGJUCUO/MGqVAlrq3dlg+OWinWeqKBw6kpIEPEJgx64+6tjV51HI+nCbHm6Vl9htggjjrbLb2aDSq1uikUnlRFjM2I6yKgWPOQfveRNcbdeKvX9cweagcNqY129q/OuUp7goZ5KUNfIA42ux/Z7cVxxVzRytvmKQlTgNlH3McXriczmQiXm9u4PP5/d4+MBoFsUEwnpTYQVSgT2UXYk8VsnThIAH+s+sbxpql563hFza27j1HnSyLNjoDHCBc/thbOy3IgDrSIL1tlkEm4pbpYDNngGCAcLucQb8fSoHRZEYgRlEAgrFBINyLAMQgs1F2mf3oOzaJkxrl31jWMEIvhgPVM+tbT7YFirZEBndBAxkggAVfHnVuO+VRSfg3T9OON8tKl6j62rMINDk4xQCRt05v6L+LFFsXgw8kXitOHKXKvyuqonXDRrzPHAEqYHO9RnTwgu/pL1sQj1dFTazZpgxkgDjREnh7pxVbULBII5M9OQws/zrC/oHDcVgBEGGVSiNTKNiwzqJMGBZryClGccorSqMqUwg2EYh9WzRB/dAiM+hUd5z2wIHK5qVZ6vo7HAMWILzBGDh4W93RoToJshIhoLsiqdlgJIABwum0gywHIRigySkWOiCAnAEIDpLQ1Dr7Xn9nMfM8jq7hN3XTFM0984xyEW/TcRcCuqBjFqXwQVvIwAQIvFC/OEyUC6mQu3KGbny9NFZ+2wM7pzgcr8fr9/nQJHhAFBEgAD0kXouh52R2EIPLD6LHFYvdoljIWz1Lv2oWcbLEHHhla1tHiGayLBzfBiZAnGzFyYUtmkiCYQHKBQnyqdAmHHYChHgjSQygQac3gs67iEx5xCOTYfqsUOcKn3alexKpPRQS/j3zTfCXBV58uM8BFxiaybJggQ9AgIBb/hvbrS3OiFkjunOuUS7hVWr3wI6KgwBESKVSgyinsDz3Vxtd1gZBHKWQUYpuINgdW10d6LngUP/wYvOS8WpfMA6u8I8POEqdcLTgFVjlDw5AgPj8EAK63YjnWTVDP75BWkFiOpIzMp6AASISjYKGE6nui5uhnuXvRbh3RcwrVTuzgRHgUjRrhMhSN2e00toRfXVr28ZjriKZhqu23yVp2EADiNNtwXd3W4PR5KwR5OQCc6WSrDIcjs/n8XX4kFkWHhAACEJyU7yLJJXigL+XXFTL6CZXUCIhUzmy1E0frrjsiLywuXVHk6d4sh8sJQ0ogICq+aft7ZccIZNKePc8k1LKL+p67POcgAHCZrOFg0GYHvR6o0AAor2iAgRRMbiEnpO+HHsanEgsASpgOFBNbJQ1tQWfXt98+JKvz6M4uB8YUADx+WHnziYPSK9vJcqFpOKMEdi/gEcrFA4qlWq5SsVkqS3mxSUqBvwIWe4cenWXAHZVyHY7fbj8sSXmkSbp0cv+/1nXfNZKqYD7MFWKPGX7UHOxbz1jCb63x+YLJ2aOVN44RQeYqOz2gSWtcDnsSEVpqq8Xi8VF84DIiI4QfLM2CAAE1TF6mlHATqQFmj9W9fCi+kateN953++/aG52USfLfJffAAEIGK7hFnXeFtIrhffMNWrl/IqH/QIg/H7YH7yJeNJADBCior/niZGS+EEMTHrOfKdwb/ex9hmWClgnJ06WyGSJDAC9PUf/TiQwQADi80MODDxWC5xkkBy9bJnsrzGJABBOuy0YCPD5PD0h2gNAFPktT445mX0KY/sscuEDaX0wVMB1UDzhQCUV8TYf97ywqZVSAeczxAMBIM60Bz/Yb3cH4rNGqq6fRIheK6tcsHJnXKSswYBfrlCq1GoEXxYdIEhKCBzTJKmRsvepTpwsBRzsLm+aStTPLw45QQUcpFTAvUmu5gECnnMgdAVGQLm4a64BPI4VVy4gcyYiq87lcoSCQbDsicXSoqMDamGOOYGGgy4nZW+zuue/RxMphYT30KL6JRPU+Pz+Phti+cDhXFhpg+SpmgeIdYddO5q80CmgXExowDqsiq02Y4Dwd3g9cHM0Gs0isbiIHtbZqQkbBLCIuFpXw5apFlYMaFA0cj6iwuFA5Q8nQAUMtzqkyabX1SRQ2wBx1hJEtlJbR2TWSCVS0YoF3GrYPrD6hctpC/j9MBPAAIFUlMXdQbDWDIaek3wg9Jz0yk8COPisV4ngHDF1mMLpj774VeuWE87qeK3k14Hy3lXDAIE05+/usR1v8RsUwjtnG+pVwuqJWkI+OJcDAOFDBkolDBBFSjOXMzcIQqBYNrsEYd8jF43HyGv1xBPJYToxMGJcvazdHXlmQ+vuMx15PTn4bqphgFh3xLXttAfKxaqZ+klD5HipVtFWkSHaCwT8BqNRKpWVKFYChk9YO8gOgrhC0HitfJcvwBUnP0hxDirg4XrJRXsI2WXwpsn3+cF0X60CxAVb6JMDjnYXo1xM0uDsqlLp5K6cLdg+wDbp9XpisZjBVC+WwDJSEuwinpR1zA6ighFptblayLFwKjV1mPzRJWZoHE3tgd990XzeRp0suw9nTQIECBHe32s7ctlnVAtvn2Vo0Ihi1WRoggHC6XTASQofdDojfChLYaHESMLDgo0fz6gYtblYK9Rq+E9BeCBSenCRSSPjH7nk//0XLW3uSIWaU6XV1iRArD/q3HzCjS3Dyum6KcOIclFsF6R+jRaxUDrsAAhwcKq12iJmkeraLIBDFiAGEX9vv8am68MsFfD1k7V3zTEh+dies16apa6beGsPIC7aw8j/AQ5eKBfXT9Ig2WT1KBescPFOd7uwg/AZjCaZjOBXiS74SYEag6gYJLE1vQqRALaeIh4XHpbws8SG7KsTxMnST7PUZWRZYwABt5YP99kOXuxAQPfqmfohOnG1ZQrCaz0cCnk87mg0YjCYYKEskX6BEUTae5L5Pp1UqpDlQZ8h0kumJAIu8gPcACrgRBIxwa9ua0c6CSocMsdqSwpfnXRvOOYC6t88XTd9hAJqZOnez4VJBivW5Xb5OjqAFGDBkEgl8HQsrKhen2Loe1kbBFUxepXWtW4ARiglvIcXmheOVyNl4Qf77IgMpnThNQYQl53hj/fbLznCM0fIQaKnEPOqx/EhO/vwUnc7nb4Or1gsUWl1TJKYUukYPC7oOQnEUxWjX/DAPAwXO62C//gS85xRKiTLf2On5fMjTpqpq2Z2EAjoBjrsPteBXIO3ztSPNIgRhdH/aVH0EvBGhwGio6NDqzfI5Yqil59bIAdRRyx/7yCm5yyWhIHiwIgGrejry8zThyngQPXylrYtJ93FKr9Gy6kZgNh9xguaA7yNb5yinT1SRQIYS/ViLnwoseWPRqJejysSCemNJqlMXsJ3O4c4dJOklIyRkpEG9aQsfOzwJEPzWTfSIEG2WyQNgAPVc5tABTyonSxrAyBaXWHE3l10hGeMUN40WQflotpOLtiJiRXrcbs7vF6sVCShlMlkJYUxgg84xcj6QVB86Bc+kIcxXtiXThoie2xp/Qi99HRr8Nn1reBw7HfBtVpADQAElAs4TSLZZKNGtHK6fqRRUrUhugAIEHkjiFMoEqs1WpFIWLojDAaPcMyZtUGkKD4UZRWydofZI5UPL64Htcqhyx1Pr2/BbqIohddcITUAEPvOeT85aMegXTdRM2+sspqTHyArrcft9Ho8Wq1OoVSUWglKE3wzNohS11VzM7s/DYYCC+xF2oj7Fhg1MgEcqJ7dMEidLKsdIGArQsgmTi6mDlWsmKYDKSv8rPsz9qV7FueasUjc63GHQkG9AS5SipJuH1g+TpZci3G1rlKxlE7gJS0ZOixSk908VXfnHINEyIUD1Uub2xy+QZfJsqoBAs4qnx6EcuGt14iADqNNEqgbVbuRhkXA63Vj+wDnDK1eL5crSwsQDCLwEfHNOErRHUTR8QJZpwANt4EKeKYe9t8vDjte32YZbJksqxogDl3yfbjfjrFZMk6zeLy6avcOaQslDxZKl8fjEglFGq1OJBGVGCBItXCFYPh7acKYouMDORNCBiolqIDngQpYh2N1zMa1e6zBSKk834rfh36XWL0AYfVGwcvc7AxPaJCtnKGrwpiLbsKHuRAe1jjFgHlSqVT1e2h6LwAzOK1ikFgMqmL0LrEC7gBGIFn+gwtNSydpsX1AlrpPDjqRAreAomrxkSoFCKA1dnQgUzSpWOVCWs3KBRl4DiceS8AAgTTW8LBGJutyOOEhnJOPtJTE1ZriQ+mWH+J9kFLgkUX188cq8d56bVsb/P0HiU5XpQABlrS1e2wYcgzJsgnqciy2/s0vRE0h/sLrxrxJanQ6YoAoS4oKZL5PU2MMkgnbv2Eq7GnsFoARI43iR5aYZ4xQYFf7/KbW7ac9hZVWW09VI0DYO2Jv7rC2OCNjzbLVswwyCb863aJyRxpbfa/X5Xa7+AKhRqOTypAkphyaKkuuxVBjVKPjeW0thmu0FvCLbLcTG2TACPx71hL644bWgxcHPhVw1QEEoBo0WTvPePRKwc3TtDi5iNYCcwE8nnF+4XY71WqNUq0pm9czoedkc1LSzPclRiP4T+E0eeZwJZg1RpnEx1v9z3zZfKp1gDtZVh1AwK31vb1WRFrMHgWaLA1hri7xwPe/eHhAgCAT+ODv6NDp9bBQ4tf+F5tPCdi5sNw5tSCnfDpU1fcwAUCpheNU98431auFBy52wIFqYDtZVhdAOH2x17dbWl3RUUbp7bNBowgO3qqeMWzj4NEInl6ccWL6qDXwoSy1B0SnTBjuHOIoVaLE2TUg/fI2kSFe4SDbALLUaWTCnWe8L2xus3oGbCbLKgIITPIvjjh3n/WqZXycXIypl9ZKxg64SCH+wuNyYsOv0milshJmkeq2HMCdQ8i1yEar+nda5V3KJasNGAEny1UzdXCgkoh4m465XtrSjhQSJauwkgVXEUCcbA28u8uK/NRINnnDZG0N7ZnhAQGAcDkdUC7UajWb5ak8F0PPCSMlYfguT420FkgAVnORgHvXXOPyyVoE4Hx22PHmTgu4/AaecKoFIMDN/fr29jZPBJRH4OCViri1YnRjTADEQolTTo3OoFIRu0l5JgpJa80YKSl/b3kEnlsLNrxIOwAHquvGa+Lx5Hu7bR/stVVnEqP+CKcqAAIrat1hB5QLuZh/y3Q93KKqhGIzH8nCAAGKPUR5JxMJDXwo1Wps+PN5sCj3sK7WqJqqGEWRZ58Kge8/Yj1x8Dl3jDoQTqzZZvn8oKP6j+T71MeqAIjT7cF3doOIPYVkk8un6EiOpDK9g/skq55vRkYGUGAAIGAOUGk0SDNXzhNHLhOsxWgY5UOlIkhtoBQBODCphY8vNU8boXAFYi9sARXwgMpSV3mAAAfBa1vbwWgEb1akHodyURMnF9kZjqwtUC6cDrtcoYATBJI8lRPe+IwnJZMrbQAqwDUBI9i9DTeIH1tKstRZPTEcfGIvXBMtz6eRlQeITw+RkwuRkItUtHCLKufqykdAvd6DBiOHNaIwEKMFgCjzOiV+ECTsEATf8drZdfUq1Fq6gaT8S6QmNEofW1I/wiC57Aj/z7qWI5cHCBVwhQHiVJv/3d0WMBHMGqEkBmGo0zU1zWGACAWCLifC++IqtQZnnOXxsMYCYrUw6DVM2to6mCGYr8p3gFJLi7jEbWXpwmeOUDy0iDhQNbUH/7Cu+ZxlIFABVxIgApHka1uJW5RZLb57nlEq5tXc9oF1kXI5bXiTAyAUihJnkeoy0QlCkGhOcqqaiiOgk8SUlngp0OKvIgGC0XV1i8ap4WRJqIAv+/6wvqXFFa51gVUSID4/ZN9zDhTYdfA5gVtU9q1YQzLl8QhAOOx2JLDGEQbotsuMcSSxdR2l56yKKQOMwBb4JiZLHZyAoTg/t6HV4attB6qKAURTewDJJr3BGNyiQHUBd5+yuQ8UcTYxBogOBHGq1Vo4WZedQ5fkpAR9DnpUQgKOIsproBcF7z6QGd02ywDeeVivNx93v7SlzVfLDlSVAYhQNIGYi2ZHuF4lxJYMWb1qyG8y5/yCGw5HcMAZj8WUKg2MlOU+Sshw50DZgIpR5s3LQF/sBfYP7zmxgItZfdNkbSyZ+uyg40/bLaGapQKuDECAQHnXGS90tttmGceapSRErqZsk+zcwfYe+oXTbgUfp1KlUqpVZUhC2W3aMvS9XEKulaRpKQtc0kV/DG5+CgnvocWm6yYSKuB3d1vf32ur8oyqVxNCBQACyTbW7rFDN5s7migXiHupxe0DAxCcoN9vt1klEinOLwQCfplxDhZJAhAMd06SZL6nV1VIgKX51MqFjy4xzxujggPVmm1tXxxyVEXj+tiIcgME9lpvbLect4bqVaK75xpVUn7Z9fY+Suiat/v9HS6XAxFaWp0+WYlEpiz7HrVBFHNQi1EW47qWMmvgQGWeMVxh8URhjNh8wlWMsstaRrkB4ssjzu1N7ngyiXQP483MyUVtplvFqzsSibqdrlg0qlAoQaVVfv2C2cWQC9YH4ihVi3paWWd7WStjMQK+f8CIiY3yi/YIstTtOVdjTpZlBYjz1uAHe+02b3TeaBLQjYDZGgrK6ja5oPmHAn6Hw4oPClgg4ENZiWgInGJw68ggxmvLQb2sS7VilREq4GTdlKFyBHTBHbvJQqiAj7fUkpNl+QACJxdv7rKeag/Uq0V3zDXqFcIaNdtkLJTcYChgs1nFYrFarROKBBV5gXfuICh3TsVw4FoVk0w+qTqY2+BkaVIKj172gQr4nLVmnCzLBxAbjrm3nvQgJ+3q2YYpQ2Tk4KIGTy6ycwE+iwG/32m3KVQqEO2VOQSjsxmwUKZtEPQUoyoRgjFYYrYsm6S5fyE50d93ruP5jW0tztpwsiwTQAAyP9xns3giC8aobpikEQl4Nc1NRHh6Y1F4QEQjYcR3EwtlJfQL1gYBSg5ipIRAaSxGlUIEFMAUUn+tmKq7a55RwON+ddKNCGZHRw04WZYDIECK9cE++7Fmv0ktum22ASoG8spV61Dm1S4SoxUM2m0W+NaCRKuCAIHmMmkp8ZpidhA0FiOvAazATaDVQMjyHbMMt882AMs/P+x4Y0d79TtZlgMgcLqz4ZgTHIerZ+qnD1ewWllNX3hth0Ihu9UqEolwxCkSCWGNqkiPCD0nS43BBmtVpBG00vwkAIyQS/j3zjeunK4PRpJr99nhQxWpbtqXkgPEOWvow32ONld0wVgoF1qxkFvTykV6JnDqgsGA3W6BfgEmzko6eiEtJbuDqGl/kvwW2AC4CzY4rVzwwELT9ZM1nkAMXtgfHbBXSD3NS5ylBQgoF58cdBy82GHWiJAjvFErgoDyalcV3wQDBIIvkMQ6HA7LYIAAQFRUY0oTfMOTstY3ZlU86EVsGpYAlsPDi80Lx6ntHdGXt7R/edRZxPKLW1RpAQLGGJB04zgTPBeI2iRmtBpXLli7IKCBGCDq6qTsDqJCrwBWlukdRNpIWdzpQUsrvgSwBIARI0AFvNg8c4Sy1R15bmPrtmqlAi4hQMAt6sP99lZXGO7oN0zWIMRtICgX0PMBEKGQ1WIRCATIMSeRVC5NHoMQfOSMIUwN1JOy+Iu5RCUCI+D4OqlR9vCS+gkN0vO20DNfthy40FGi6vpTbKkAAgQBnx1y7j0H5UKM2PjhBskAUC5YQSN8MhwOYQchk8mxfajopihFjJQkLSWlxujPKqjAs0itiJ317JGKhxbWw8kSDoRwoAJ3VAWacs0qSwUQO057Pj0E60vqpinaeaNBZjsAdAsiSGKAiMfB4h0KBSVSmd5grJR+kR5WljuHUGNQ9r1qW1y9tAcDBuPV4gnqBxYQJ8vDl/zIiH3BFqqqbpQEIMB3/P4+G04u5owiAd3k5KK2ctFefYgQeREJhx02KzYO2EFUHiAyNghCjTFAQLiqFkhpG8OiOrLUgcVPKebtavK+uKW13RMtba19Kb34AAEnqHWHnTubvEjvC9vkKKMkWt0nvX0RF8kBEYlGrJY2mAYRogWMqPiyJHlroWKQTVrNnxD1aSwGxs1svCKCm+FDKBRyNxx1rdna7glUi+N88QECqaI+3O/AVhzxmjjIgaJV+wcXnVOR2UGE7JZ22Cb1BlMlPSAyjUq7WtMdRM0CBjCCzyVZ6lZO08HC9cF+21s7CRdENXSoyADR7Ayv3WsDTdb04fKbptV2QPeVwwPUSyTi4MgJBAISqdxgNFWc8A5RQNhBMATClJ6zGhZUgW2ADi4Rch9cVL9skhZ7QWRsZbLUVX5LWEyAwGYBHubbTnnq1QIoF+DgHTAnF+ywYx1GoxGb1QK1AjsIncFUcf0CjWL9IChAFLg0q+YxOAEgwdqjS0wLxqrhYfjylrbPDjkqvkUtGkBAj9h71vv+Xjv8iJZM0Cwer0bwRdUIvzgNYQAiCgME+qhUqkCTUwUAQY450T24WldDY4oj6MFaCjDCpBKBChgkXR2hBLLLbDxW4Sx1RQOIdnfknd0koHtyowyOD2IBrxZ5Lq49M4kBIkJ2ECKJWG804eZqWJOsoxROMRIVcugcrMu5JP3GNpyhAjaD7BOJnf/wZYWpgIsDEEC+Lw47dzS5DUrhzVAu6geacsHOBSxCX4fH5+sQiyXEAFEd5leGnhP8nOmAzpJMW1poGSWA1TRpiOzhRQ1wL4Qj9u/XNR+tHBVwcQDiwHnfe7ttSMAOzWLZRE00NtCUC2KA4HLj8Si2D3hTE4Aw1VcDQAAaWBUDGboYLaOME5lWVRoJYAyxj5g3hslSpxKdaQ8CI85WiAq4CADR4oy8scti8UbGmSWrZujhFlVxy0opBg7Z5WOxmKW1Ff/Ch1KhVLMHixVn/SGu1gzDN05YCFAwF/6TbVu3D6UQDi2zuBLAkOEIA+/a+xeYVFLBkUu+p79saXVFiltLPqX1ASCQ2cLWEbW4IzA0wOKQZSU91uLfdtqtkfNvma4HB+8AO7nIChEcuaFQGE7W06ZMnDF1klLKA3uSUsKTiXkCPocsyHzkXYJ7GEcp8Pd2poTAZ9ARoW1oZLcfqQhoQuPCSzAMxS6SyZpft2Ka9u65BrGIu+OMB0GfzrJTAfOefPLJPLt2tNn/20+b39ltXXfYBe68dUeciC3RKwQyMb+pPYjM//ctgN2OLJWBeaVSoYBPJORNm7NErjOHYym4soAHCLkGdXKBkM8BMuLYurw5ncjWIeD3nT5xFBWPGTsB/KAQPhoDxxtvKBYIJ4PRzh9/OAE/HBGfO/DsxwNyymEpwXN3LHnppk63Bc/ZQ7F4cuowhZDfh/d6PyVDmOPzLAJHmD96tem6CeohOhEeCcVSBy90GJXCf3t4DHYT8PQYZ5YOjIDuKwVCYrQS8VTEL5Ypn9tsO3TRm/Yf59QpxLw5o5X3zDXB+8PpAz0FSWFctgsNs7S3rn3rdfhZr7zt7lFjxqFytYyPLMH/+eElf5iAVnaAA5H4HbON31hmxjAFokkmGza9ql0CoAsPRRIvf9WG3Eugknl0sfnry8xFwQhMA7z1kV8b+80JDbIRRsmVsujDDuJ0e3Dvee9TXx8PvQghWLdM041rkL2102pQCZEtSi7i1TTPxbWnCRPEGYsFvXUixa8/udTmiiDDBfICNahF4Why7V47hDN3lFIr58PFBQuybEsPDQsGAqdOHgNAjBw9Dulz8c4BZh29HPjlxxeH6cU4USftZH5g8Zo0RA41ECiGwSonkFX7Kqzi9uENjmy3o41SdyDe1B4AbSXMfBjHfg4f+Ht+9fFl5Lzbd75jR5N34zF3JJaa2CjDjjhXGH0BiLbglpPu22YZTCohW4RUyAOqDdWLkcw+EBmQpsm0rOAZhQz3bZfPKbX1XxxxgW3xXx8ac91EzdKJmtWz9LC/IMAGVkxYlYDK+Onn4OU/XQnBVzB4+uQxWChHjBqD6FIoETA0XLCHkaToR6uHf/OGRrad+EF0DHZ5gDBoRmQWlA3G8u8PvbMnCeCdIxfzRxrF9o5YkyWE7aFKwh/fICtYWiDQ/vs3ziK0/L4FxjtmG2aPUrqCcYAFDGp4o+QW2wdlBlY6+PsDeI41B442B/ac9f5xYwu2qVA6oF/ULoler1JmrHopH66ONLFilHGS90cSyDsK08P9803Lp2g3HnNiZyEV9UGkvVbd6w1oGxf2Bi5pYW7eWrL2U3VXZkxGIp9AFMehFB16FW113QDdtVEjAl349GEKkFeCCnjj8QKdLDFjn9/c0uII/8cjY7+9fAheG9AA/v3hMUsnqNdsawcGFQgQIIHEy+dXH1964uXT+Pnrl5t++1kz3kiAnNRASfdwlUnBxdqzWy34l1gFMxe2ZDAB+kIJaInThilgYUYiDEEZDUhsQxDNCWcIfAB3TrZtAG4oESCDPHzJt/98B36wkzxwscMbiEMZpNaH6lr9ebQGkI58z6PrJY8tqR9rll52Rl7YVCAVMKjGNx93r5yhByFgtmaYrn+8esQPbx2OI7kCAQIqhEDAvXuu8VvLG7+3YsiPVg/7zs1DLjvDr2+zwMiBlJP5WjvzEEdV3QJMgIXSZm3vZtAlb29O+tRGLuEBJL2hOKTLMF2V7wJJBxQNkuYQZlRSLRlgaJI4ZFmzvf1HrzT979fIzxOvNP3962eBFzICEAP3sKl8gi93TWwGqmnD5Y8tbRimJQ5UoAuHsamv7cCa9YeTs0Z2USVQCPLo3jZLr5YSz/3ON03+pWMBiPgcnGXCSImNAzTbJ1YNBwfvC5tbPcE4jtbyL6q27iTB1PGExdIGV+vcHURuL1gDbdq2XF6khIsnzB/QKJLgzslUDTsIwodvnab/q1uH/XDVcPz8ePWw768cNqFRFoIZNUWSWdKr5iSA82ksw/mjlaAL1ylABex/flPLmT46WUIvxnxRywT5dL9vLzu8pgLhuNsfa3GFvcG4XMyDbRxhZ9hpIxY6n/pq854UeHo7vJ5ujWe8FVPslqzVGZYIePUqUfnzthAVg0uOq+M53DkEzQVcGCbvnGNgf5DUDO8H5D70hQZcmG1tzqrCWs0a+5ZOVD+4yASb4p5zXtgj+uRkKRHy4AAMa1Q+DegbQKDMjlASS2L6cHKkB18p0F5MbJDCU4gxfA3Ai0kSk3TYLfCw7tY9hYiH8wvYky87IptPuCcPxQmihLyfyysGuFrzOAx/LwGIVPZsAmgejhKrBIxS2R9Ea0DByNv3pbw9obXlJwGcCWAQkXLlvvkm6JJfnXC/8lWb05dvJsshWjFeHjgu7VYbVI8XN7d1C/roA0CgTXC8+euXT935i8OP/ffR+/7ryD+8eRaV/eg2YtgIxxIDcgvBAAT0CwuSR7H6BdZ/KJLsCMUD0TgYFmH8+79vn3X6Yw8tNEHuOEEs8/EhDBAwQzAA0SWRIfRV2ER84Tiayv5g0xeJp7DTgT5IMSK/xVild0GFFPF5ILu9c7YBnz8/5Hh9uxWesvk0t0ErGlsv23TchfmQe/+XR5xIq23tmjK3D56UbZ7InjMdLn86VBNqTINGCMdPsHVj8lU5B2k+guvxHqw9eECsfedP7a3NSxbOm7no5m8/d+rwRe8QvYQ5/ayLJlI6Of/bNw1ZPdMAIWCQyqlsEQeNaOTdN16zWdqmzpy97MZb+HyeXs6HE8Qj/30UHlM6uTDroILmzRih+OkdI5FxAGcu5WxnwfKnD15DArB5YeMAU/SnBxwKMf+x68w4B8WJwbWFhjfcJ/vtf//mWSie/+uWYfUqIdSWL4+4nnznHExUv/uzCTAdZEvIFyBOtvotnphZLTJrhNnnYQzHbgc67QD2ocQK9Lidb655CXR7y29cPn/hwkMX/W3uMELaias8p06nFMBNtVEjjsQS5WdzZwAi+v47f2pruTx56owbV6yCwiEXcbC12d7kgZcncX7PTBiogUaVEIdbQgHXHyorkNF1XgoJYGBxrI7ISThiI/cUzJZ/fkMD9I5eoR8bSZySIvoLZxZ4WyBm53R7AExfP3tgFDYXuU3NCyCCkcR/fHgB2wc4TX7/1mHDdGJiimMCikli/4HrdAOdApuE82dPf/rhe3y+4Pa7HxgzZqRMmKOXYfVx6mDtCYQTZXayZkeRBYiP33/78sXzEyZNufnWO/ANWi0T8qDvXDEpyWEHRhN6UJkNJaVYHrRMSADTDwr+BSsxH+w844Hv/7dvxmZW36twYJYCVc+OJk+rOwwnCDjyLJuoNqpImFXulZer9Qf7nR/vt3tC8ZXT9dOHK5jIJXLcgh/MuAFpemBED9Iqkqyp6dTxluZLyEA5d8FiLk8AIMBeHf+yPzjBQSwNywxUflGQI9hE4vzZJo/bhUCMseMnMj4RJNQCe5xsI9Mf4sTJOgYSz16nD72hdiSAqQdvAwTatHvC56yhS/Yw9okjDD1EXnVZ+VxOo1YEaitkkIXLAtY1wrKv7HTPRkpsqvfu3rV27dov132+Ycfhj/e2eCN1i8Zr4FAMV2Jyig6/60qsh7KNGjbqhIMz6Hc7rBfOnoxGQjqDQSKVogEAR5xUZH8qmysW+wWBUAj2vST8KGNRtJpNUUkyjuQ0Mv0ZcSIDOWKmbLOjuipiDGFJZKl7ZFE9HJQuOELPb24FLW4+rcQhCBzncPB5tZu7qxiwhG/etOnXv3t29/HmmFDDTUYEyaC0cebE6+79u0fmT2qQIDPqgH8BYY1FIuFD+/Z8/Mknx46fbG1t0ajk99x916Nf/5ZCqbryvDOfkSjFPUAxjFfzxQtvrHn52LGjQ4YOe/ChR8dPnIx0VzTJdSkEXs1lwhyGoJytp10vbWlHBPeskcofrBw6eYictJnJfIatZQHt7wIQKGftu2/99F9+d1E0WzrmBq5AjG+S3pbQsbeWTNQ/9ct/Hz5sWDQWvXbupFrfvuKdnIjFXn35uf9+7g2HZKLANLWOy4s7Tkud+795/4rv/eAJmVyeGxZVgNCL8gg5XgmHPnz/7WdeWnPGGquTGuuiQSXHdfsN87/9ve+NHDW2eoCsKP2lhfQqAfj4Y9uL08pXtrYjNcmyydrH50uT7ssnm87H4vFhQxsnTpxoNJl7LSf3hi4AcfrUie898X822ofrZt/PxPOwi50T99vDu576p+/e/Y1v/oVILO4eNXjtCpm44nyU8857sp+6gk2h0EOe6/Ls1QticsDy31jzys+eesHVeLdsyAziyExSRteFbecEp1/78ddWfu+vngBu9pxo54qSC20zg/tZwV5RClwn4ff98vPP/vLZt31DVkqGzubwhHhRRL1t0ZPvLx/D/c9f/rLe3FA0ILt6N/rTwT7NVHpzPhJARCVSBH18wP7OXqftwvG685+2Xz7rTqrqoInGnIsmN/74B39xw403kfjf/K4uAPHH//nNT57elJr6TbHSkEpkHbM4KS4vdP6ribEdP/2//8/UODKGTQQxQJD/sfWwAMCmGGCsE5lvMr+wS77n29h7Ov/aGe6QAytdYiDYijL/dHY0e1Mu1nTDphysynzMNAAFQXCRkP8vv/HovuQ85cSVJKaJJZsgRhdex4XdI21vv/raGoOpMb32ujYjt/BuuJhuBjlG6IKXGYGle9F5W46Jhyk25y+g2+PxTh07+uff/atm493yEfNIOxnFj8Plh70WzvHXv3PbxL/9x5/H43FiTGY6eI3Dph6gOWf29ADumfYwf0q/RXqcb9co+SpvgZ6K6QmE+ghM6du7/ievJZJvRcV4meX5Grta/lP2cYTupbiC376+/qn/+HlIZJZNu48nM+D7ZCQQOreh0bfjD7/8p5tXYG7n89YmyU47W/XkP/zdP3/QrF30XXJyl0sVzeEmOtqCO/5r8m1PKBompeLRNAykV3W6qq7rn1kg6aWfXs5X3MAulfT8z8JEdsHiG/I3Zg1hR8OgHrug0tjEhi2ng5eZ2Mr0I0zd7OecD+kmMUWiRNbOSlQz5k5yP5fHt5/b/+lbz0fGPSw2jU/FO/MIc3iCKFl7r6xevWrM7BUkcIvodWw1aZMt2wCgJlMtUztbV6afnY3JQGnnbcx96CK7+jINY8pji2O6Q25A7AWXs2ntC+9uOpGa+JBApq5LZlziyB0C/9ktE4Ib/v4XL/CFEsIe2imzHIxmW8hMqey4MEFf2QZkgDGL+l1wP92eTAnZgez2eLrAK+dGdtTYZcqUk56y2amb+T3nrzlonm1q55OZ+ZEuM2cJdC0qp8bcijN979ak3KXUvW2dAsxpPAPGnZV3b3Pn72np5XSgS11sM6/V5ZweZkYHLzmEff7dj/76xW125bxv8SUqhPGxYwlypcDxjxbKT//p5WdM9Q1sN699dQGIf37yH37+3mX1wu8gvrlL5mMAhLctvvc3ix/4ibJxAlR0FJrduOPlxdaRhZrODznUU+z97K2Zz1c8mP4bextzV/ZT7uOZCnLLZG/OPpPTvEwxiGAkb9pOgeTu4dk/AAUcJza2ntgmmPKQQD0sZxuFZcpPhNyRI2/qzUPMc+8lb2amGd3eHN2+yq0i3Q6MaZc2dGlSzrTKmTWZpZPeBTCYdGndr8LSYZJxt9QR5aIz8IbDl4TbDsWOvDJ61d+I1WbiIc70uBOkMgLoEZ3ZP2b3LLkzuBuaMPelgTXzCPtdl+dznur6JmAfzzSNwb5MG9OQnUHwNNSyqJnG3GzVBGjYvzAfSCGZkjPvgM6XRPoutpAM7BLMZcrNfJmWFtty/JWtgn2WfUuxd7JfMY+n/5ppXua10fmGyKI005gsNMNqkBZYusw0kjI9zQg4U3m2hWyF7JOd48W+KblCkejyqf1P/+6ps7Kl6vE3pCI58eCwWwW8dTv/37tP/9Py5Su6TrZMWV3/2wUg1rz83BNPfRwe/zWJrjEVC2drxxT0nfxsseL0T//xn9XGhjhDjMOuBGaZpBcFw7+Q/oX9gB+SNDW9BBhbatqkSpyLyG8ZXonsUme4Jpi/ZUrAJ+Jzkf6yswSmtLS/Vvoziybsg0z1pAEZzEhjDXMPexLD3kY+kcyu5Cts3dvOHFzz3G+D4x6X1k9MxTsDWqDkRzosdQefufve+0bOugn0OWxNbDczja4j3vBpnGLFk/srW10WxDK/dt7PtD/9/4x8mGHoApTMHNv99n+e9cuF426HLRmkOdlhBUAEm/eJml5b+Zf/IVQYUkmmRWnJsOVnGsGWzIosWwsDQpnfWSjrBMLO7nRF+ewrgeT1zgxW5yuj88XQteoMUGZnCCud9NfZpzoFxiBhpnnZicd0oBN3cwSYkQrbjew/OSJl78g0pPOmzm8yEutyW3ZYs3LPDhwRYFcLVddXQM4TV//I9qbL/uBaz+VsBjlcocR59DNr027BhLuEhnG5u2CACsifYnt+99sf3fmNrz/OYUJ4rn116crlixd+8L///sPTYt3Cb/KEkrpEjNnU8oLtp1MH/+eZn3/30UcfwUu2tzJr++/IbX/PA4986RyvnnEPuppKMkGcECVX4Dm9cWrgs0/ff9NUb87MzkzmlczSyY2h7bIUu87R7IpiZlN63aYBK4MiBLOYnNTZnRFBVXIzUUU+fu/tv/nFq5Ep35JqG7OTADYIYFLH0bU368/95tlXya+ZFPcEzdjSWNBk4TVbdRq7cyCerTezOLPg3r0xDBywSMkWzhpwuzyYgRkGhzvvYZdc9v5MTxlsZ14I6f6yb5EMsGYaQL7JvDa6pDQjLy1Gooz5iPzCDgrTqk6wYatjvs/Ce+aGzPq/Uj7MmKRL6/5sToHp6nJgNDvi2UrZdZIBxwwoZ/CIRajMtEo3iJVwerp0/Wu2O9Armw+v27t1fWT4KrEJBoHsm57sOkju5V1PPfsPDz/00IP5HHx294NY9/knf/Oz/zrc0SgYtkAo1yUT0bDjgtSy5ZsrJ//8Z/+k1uhqe/Xn1/oPP1j7xM9+f166WDPueoFMBbnGI37v+d2atk/+7YmHvvUX386vmNLe5XY5Hv/GX356Wa+acb9QrqnDToHxoXSf2WqyffL7J//XnXffW9oWDMLSWVTNdpz8yr7lu32Zxp3sroZd5+zVuUdmH2MQhwWvNCikATEDi1lIZQpJg2zO9opFzyyqcvkCa8vFn/z0b7f5Jxlm3JWKQsXI1M0TRNwtkkNPfbrmN7Nmz81nAHuIxdi04cun//jC8ZZgkKfiJWMGQceqZXO/9/3vw5M3nxIHxj0vvfjc069/ftStCnLVEK8o5R8nc3xt9bwf/ujHrKtiNVx7du38x3/91fqLwpR+qlCuB5jHHWeG1Z3960du+OEPn4DBtRoaSdtQEQn8+hf//n+eWZea+g1V4xSyCwaAcPlRv8O75/k/v07/1K/+UybvnnKux3b2HKyF9M2HDuyzWttFQtHY8ZMmTJpckU5WttID+/Z8+snH5y+1wpl6qNl4y4qbl16/vLJNurL2k8ePPv/iiwfP2N1hnoiXHKHl3nvbivsefDht46q25tL2lEsCVqvlX/75X5755FisfrHEOBrmhoinLdm6e/mY5H//8t/yX9F5RXOWq1O0ngIlcPH82baWZrlCMWbseGl+b4YCa6KP1Y4EkOXoxeeeXbf9oC0kSnG4Kn543gTzd7773YmTpuTfCQoQ+cuK3kklUHsSOHfm9KUL5xLJhMnUMHnadDgK96kPFCD6JC56M5XA4JJAvi7Zg0sqtLdUAlQCjAQoQNCJQCVAJXBVCVCAoJODSoBKgAIEnQNUAlQCfZcA3UH0XWb0CSqBQSMBChCDZqhpR6kE+i4BChB9lxl9gkpg0EiAAsSgGWraUSqBvkuAAkTfZUafoBIYNBKgADFohpp2lEqg7xL4/505qbwetZuaAAAAAElFTkSuQmCC)
Given, ABC is a equilateral triangle where AB = BC = AC and BD =
BC
Draw AE perpendicular BC
⇒ Δ ABE ≅ Δ ACE
∴ BE = EC = ![](data:image/png;base64,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)
⇒ Now in Δ ABE, AB2 = BE2 + AE2
⇒ also AD2 = AE2 + DE2
∴ AB2 –AD2 = BE2 – DE2
= BE2 – (BE-BD)2
= (
)2 – ![](data:image/png;base64,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)
= (
)2 – ![](data:image/png;base64,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)
AB2-AD2 = 2![](data:image/png;base64,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)
Or 7 AB2 = 9 AD2
Hence, proved
Question 11.In the given figure, ABC is a triangle right angled at B. D and E are points on BC trisect it.
Prove that 8AE2 = 3AC2 + 5AD2.
![](data:image/png;base64,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)
Answer:Given, ABC triangle
Need to prove 8AE2 = 3AC2 + 5AD2
⇒ In Δ ABD, ∠B = 90°
∴ AC2 = AB2 + BC2 ….eq(1)
⇒ Similarly, AE2 = AB2 + BE2 …….eq(2)
⇒ And AD2 = AB2 + BD2 …….eq(3)
⇒ Form eq(1)
⇒ 3AC2 = 3AB2 + 3 BC2 …eq(4)
⇒ From eq(2)
⇒ 5AD2 = 5AB2 + 5BD2 ……eq(5)
Adding equation (4) and (5)
3AC2 + 5AD2 = 8AB2 + 3 BC2 + 5BD2
= 8AB2 + 3
+ 5![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACYAAAAkCAMAAADSK7iXAAAAAXNSR0IArs4c6QAAAGxQTFRFAAAAAAAAAAA6AABmADqQAGa2OgA6OgBmOjpmOjqQOma2OpDbZgAAZgA6ZjoAZjqQZpDbZrb/kDoAkGaQkLbbkNv/tmYAtmY6tpA6ttv/tv//25A627Zm29u22////7Zm/7aQ/9uQ//+2///becqOcQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAA8UlEQVQ4T92UyxaCMAxEE3xLfSsqRZHy//9o0hasGqgLFx67yqGXTKZlAPjOuiwR59FWZrGHIjlEOQKq8UeYTok1ClHqmiEi70NhRzNq1KFsFGPaGfAYN8Vk7bojDnKuCCuJKjePbnp4hYJ3uahmV4e5lwIsS6E+sT4V2lpzon6Foo0lbipj1SSHGx8QFfWOJGVMk55RNASPZlbMCaL1lkY7khgXYA9UwjQbmp6J4IKtyaLCEUtOfxiz/sL1t06lOHRe1nPEu7CXiPddfRDxPsx9kT6a75HxWy7iMcxHPII1ERex9lfTRpyx9lcjHWnHszudiRbYZYw7bgAAAABJRU5ErkJggg==)
= 8(AB2 + BE2)
= 8AE2
Hence, proved
Question 12.ABC is an isosceles triangle right angled at B. Similar triangles ACD and ABE are constructed on sides AC and AB. Find the ratio between the areas of ΔABE and ΔACD.
![](data:image/png;base64,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)
Answer:Given, ABC is an isosceles triangle in which ∠B = 90°
Need to find the ratio between the areas of Δ ABE and Δ ACD
⇒ AB = BC
⇒ By Pythagoras theorem, we have AC2 = AB2 + BC2
⇒ since AB = BC
⇒ AC2 = AB2 + AB2
⇒ AC2 = 2 AB2 …..eq(1)
⇒ it is also given that Δ ABE ∼ Δ ACD
(ratio of areas of similar triangles is equal to ratio of squares of their corresponding sides)
⇒
= ![](data:image/png;base64,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)
⇒
=
from 1
⇒
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAeCAMAAADXVfSyAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bgWo6oAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAV0lEQVQYV42NSxKAIAxDU/8Kiogi3P+itjDiTs3m9ZNJgKK4NjJvnUoE7DujqQ9xEWt4Yj4mcbN++4txV0Q9EKYZrlrS+Wwzba53/JYtwzO8RhilTt8xFwhjAmy8FURWAAAAAElFTkSuQmCC)
∴ ar(Δ ABC):ar(Δ ACD) = 1:2
Hence the ratio is 1:2
Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.
Answer:
Need to prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals
ABCD is a rhombus in which diagonals AC and BD intersect at point O.
We need to prove AB2 + BC2 + CD2 + DA2 = AC2 + DB2
⇒ In Δ AOB; AB2 = AO2 + BO2
⇒ In Δ BOC; BC2 = CO2 + BO2
⇒ In Δ COD; CD2 = DO2 + CO2
⇒ In Δ AOD; AD2 = DO2 + AO2
⇒ Adding the above 4 equations we get
⇒ AB2 + BC2 + CD2 + DA2 = AO2 + BO2 + CO2 + BO2 + DO2 + CO2 + DO2 + AO2
⇒ = 2(AO2 + BO2 + CO2 + DO2)
Since, AO2 = CO2 and BO2 = DO2
= 2(2 AO2 + 2 BO2)
= 4(AO2 + BO2) ……eq(1)
Now, let us take the sum of squares of diagonals
⇒ AC2 + DB2 = (AO + CO)2 + (DO+ BO)2
= (2AO)2 + (2DO)2
= 4 AO2 + 4 BO2 ……eq(2)
From eq(1) and eq(2) we get
⇒ AB2 + BC2 + CD2 + DA2 = AC2 + DB2
Hence, proved
Question 2.
ABC is a right triangle right angled at B. Let D and E be any points on AB and BC respectively.
Prove that AE2 + CD2 = AC2 + DE2 .
Answer:
Given, ABC as a right angled triangle
Need to prove that AE2 + CD2 = AC2 + DE2
⇒ In right angled triangle ABC and DBC, we have
⇒ AE2 = AB2 + BE2 ………..eq(1)
⇒ DC2 = DB2 + BC2 ……….eq(2)
⇒ Adding equation 1 and 2 we have
⇒ AE2 + DC2 = AB2 + BE2 + DB2 + BC2
= (AB2 + BC2) + (BE2 + DB2)
⇒ Since AB2 + BC2 = AC2 in right angled triangle ABC
∴ AC2 + DE2
Hence proved
Question 3.
Prove that three times the square of any side of an equilateral triangle is equal to four times the square of the altitude.
Answer:
Given, an equilateral triangle ABC, in which AD perpendicular BC
Need to prove that 3 AB2 = 4AD2
⇒ Let AB = BC = CA = a
⇒ In Δ ABD and Δ ACD
⇒ AB = AC, AD = AD and ∠ ADB = ∠ ADC
∴ Δ ABD ≅ Δ ACD
∴ BD = CD =
⇒ Now, in Δ ABD, ∠ D = 90°
∴ AB2 = BD2 + AD2
⇒ AB2 = + AD2
= + AD2
3AB2 = 4 AD2
Question 4.
PQR is a triangle right angled at P and M is a point on QR such that PM ⊥ QR. Show that PM2 = QM.MR.
Answer:
⇒ Let ∠MPR = x
⇒ In Δ MPR, ∠MRP = 180-90-x
⇒ ∠MRP = 90-x
Similarly in Δ MPQ,
∠MPQ = 90-∠MPR = 90-x
⇒ ∠MQP = 180-90-(90-x)
⇒ ∠MQP = x
In Δ QMP and Δ PMR
⇒ ∠MPQ = ∠MRP
⇒ ∠PMQ = ∠RMP
⇒ ∠MQP = ∠MPR
⇒ Δ QMP ∼ Δ PMR
⇒ =
⇒ PM2 = MR × QM
Hence proved
Question 5.
ABD is a triangle right angled at A and AC ⊥ BD
Show that
i. AB2 = BC . BD
ii. AC2 = BC . DC
iii. AD2 = BD . CD
Answer:
Given, ABCD is a right angled triangle and AC is perpendicular to BD
(i) consider two triangles ACB and DAB
⇒ We have ∠ ABC = ∠ DBC
⇒ ∠ ACB = ∠ DAB
⇒ ∠ CAB = ∠ ADB
∴ they are similar and corresponding sides must be proportional
i.e, ∠ ADC = ∠ ADB
⇒
∴ AB2 = BC × CD
(ii) ∠ BDA = ∠ BDC = 90°
⇒ ∠ 3 = ∠ 2 = 90° ∠ 1
⇒ ∠ 2 + ∠ 4 = 90° ∠ 2
⇒ From AAA criterion of similarity we have in two triangles if the angles are equal, then sides opposite to the equal angles are in the same ratio (or proportional) and hence the triangles are similar
Their corresponding sides must be proportional
⇒
⇒
⇒ = BC × DC
(iii) In two triangles ADB and ABC we have
∠ ADC = ∠ ADB
⇒ ∠ DCA = ∠ DAB
⇒ ∠ DAC = ∠ DBA
⇒ ∠ DCA = ∠ DAB
⇒ Triangle ADB and ABC are similar and so their corresponding sides must be proportion.
⇒ =
=
⇒ =
⇒ AD2 = DB × DC
Hence proved
Question 6.
ABC is an isosceles triangle right angled at C. Prove that AB2 = 2AC2.
Answer:
Since the triangle is right angled at C
∴ the side AB is hypotenuse.
⇒ Let the base of the triangle be AC and the altitude be BC.
⇒ Applying the Pythagorean theorem
⇒ HYP2 = Base2 + Alt2
⇒ AB2 = AC2 + BC2
Since the triangle is isosceles triangle two of the sides shall be equal
∴ AC = BC
Thus AB2 = AC2 + BC2
AB2 = 2AC2
Hence, proved
Question 7.
‘O’ is any point in the interior of a triangle ABC.
OD ⊥ BC, OE ⊥ AC and OF ⊥ AB, show that
i. OA2 + OB2 + OC2 – OD2 – OE2 – OF2 = AF2 + BD2 + CE2
ii. AF2 + BD2 + CE2 = AE2 + CD2 + BF2.
Answer:
Given, Δ ABC, OD ⊥ BC, OE ⊥ AC and OF ⊥ AB,
Need to prove OA2 + OB2 + OC2-OD2-OE2-OF2 = AF2 + BD2 + CE2
⇒ Join point O to A,B and C
(i) ∠AFO = 90°
AO2 = AF2 + OF2
⇒ AF2 = AO2 - OF2 …….eq(1)
Similarly BD2 = BO2-OD2 …..eq(2)
⇒ CE2 = CO2-OE2 …..eq(3)
Adding eq(1), (2) and (3) we get
⇒ AF2 + BD2 + CE2 = OA2 + OB2 + OC2-OD2-OE2-OF2
(ii) AF2 + BD2 + CE2 = (AO2-OE2) + ( BO2-OF2) + ( CO2-OD2)
= AE2 + CD2 + BF2
Hence, proved
Question 8.
A wire attached to vertically pole of height 18m is24m long and has a stake attached to other end. How far from the base of the pole should the stake be driven so that the wire will be taut?
Answer:
Given, height of a pole is 18 and wire attached is 24m
Need to find the distance from the base to keep wire taut
⇒ Let AB be a wire and pole be BC
⇒ to keep the wire taut let it be fixed at A
⇒ AB2 = AC2 + BC2
⇒ 242 = AC2 + 182
⇒ AC2 = 242 + 182
⇒ AC2 = 576-324
⇒ = 252
⇒ AC = √ 252
= √(36 × 7)
= 6√ 7
Hence, the stake may be placed at a distance of 6√ 7m the base of pole
Question 9.
Two poles of heights 6m and 11m stand on a plane ground. If the distance between the feet of the pole is 12m find the distance between their tops.
Answer:
Given, BC = 6m, AD = 11m, BC = ED
And AE = AD-ED = 11-6 = 5m
BE = CD = 12m
Need to find AB
⇒ Now, In Δ ABE, ∠E = 90°
⇒ AB2 = AE2 + BE2
⇒ AB2 = 52 + 122 = 169
⇒ AB2 = 169
⇒ AB = 13m
The distance between their tops is 13m
Question 10.
In an equilateral triangle ABC, D is on a side BC such that Prove that 9AD2 = 7AB2.
Answer:
Given, ABC is a equilateral triangle where AB = BC = AC and BD = BC
Draw AE perpendicular BC
⇒ Δ ABE ≅ Δ ACE
∴ BE = EC =
⇒ Now in Δ ABE, AB2 = BE2 + AE2
⇒ also AD2 = AE2 + DE2
∴ AB2 –AD2 = BE2 – DE2
= BE2 – (BE-BD)2
= ()2 –
= ()2 –
AB2-AD2 = 2
Or 7 AB2 = 9 AD2
Hence, proved
Question 11.
In the given figure, ABC is a triangle right angled at B. D and E are points on BC trisect it.
Prove that 8AE2 = 3AC2 + 5AD2.
Answer:
Given, ABC triangle
Need to prove 8AE2 = 3AC2 + 5AD2
⇒ In Δ ABD, ∠B = 90°
∴ AC2 = AB2 + BC2 ….eq(1)
⇒ Similarly, AE2 = AB2 + BE2 …….eq(2)
⇒ And AD2 = AB2 + BD2 …….eq(3)
⇒ Form eq(1)
⇒ 3AC2 = 3AB2 + 3 BC2 …eq(4)
⇒ From eq(2)
⇒ 5AD2 = 5AB2 + 5BD2 ……eq(5)
Adding equation (4) and (5)
3AC2 + 5AD2 = 8AB2 + 3 BC2 + 5BD2
= 8AB2 + 3 + 5
= 8(AB2 + BE2)
= 8AE2
Hence, proved
Question 12.
ABC is an isosceles triangle right angled at B. Similar triangles ACD and ABE are constructed on sides AC and AB. Find the ratio between the areas of ΔABE and ΔACD.
Answer:
Given, ABC is an isosceles triangle in which ∠B = 90°
Need to find the ratio between the areas of Δ ABE and Δ ACD
⇒ AB = BC
⇒ By Pythagoras theorem, we have AC2 = AB2 + BC2
⇒ since AB = BC
⇒ AC2 = AB2 + AB2
⇒ AC2 = 2 AB2 …..eq(1)
⇒ it is also given that Δ ABE ∼ Δ ACD
(ratio of areas of similar triangles is equal to ratio of squares of their corresponding sides)
⇒ =
⇒ =
from 1
⇒ =
∴ ar(Δ ABC):ar(Δ ACD) = 1:2
Hence the ratio is 1:2