Class 10th Mathematics AP Board Solution
Exercise 9.1- Fill in the blanks i. A tangent to a circle intersects it in …………….. point (s).…
- A tangent PQ at a point P of a circle of radius 5 cm meets a line through the…
- Draw a circle and two lines parallel to a given line such that one is a tangent…
- Calculate the length of tangent from a point 15 cm. away from the circle of a…
- Prove that the tangents to a circle at the end points of a diameter are…
Exercise 9.2- The angle between a tangent to a circle and the radius drawn at the point of…
- From a point Q, the length of the tangent to a circle is 24 cm. and the…
- If AP and AQ are the two tangents a circle with centre O so that ∠POQ = 110°,…
- If tangents PA and PB from a point P to a circle with centre O are inclined to…
- In the figure XY and X^1 Y^1 are two parallel tangents to a circle with centre…
- Two concentric circles of radii 5 cm and 3 cm are drawn. Find the length of the…
- Prove that the parallelogram circumscribing a circle is a rhombus.…
- A triangle ABC is drawn to circumscribe a circle of radius 3 cm. such that the…
- Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct…
- Construct a tangent to a circle of radius 4 cm from a point on the concentric…
- Draw a circle with the help of a bangle, Take a point outside the circle.…
- In a right triangle ABC, a circle with a side AB as diameter is drawn to…
- Draw a tangent to a given circle with center O from a point ‘R’ outside the…
Exercise 9.3- A chord of a circle of radius 10 cm. subtends a right angle at the centre. Find…
- A chord of a circle of radius 12 cm. subtends an angle of 120o at the centre.…
- A car has two wipers which do not overlap. Each wiper has a blade of length 25…
- Find the area of the shaded region in figure, where ABCD is a square of side 10…
- Find the area of the shaded region in figure, if ABCD is a square of side 7 cm.…
- In figure, OACB is a quadrant of a circle with centre O and radius 3.5 cm. If OD…
- AB and CD are respectively arcs of two concentric circles of radii 21 cm. and 7…
- Calculate the area of the designed region in figure, common between the two…
- Fill in the blanks i. A tangent to a circle intersects it in …………….. point (s).…
- A tangent PQ at a point P of a circle of radius 5 cm meets a line through the…
- Draw a circle and two lines parallel to a given line such that one is a tangent…
- Calculate the length of tangent from a point 15 cm. away from the circle of a…
- Prove that the tangents to a circle at the end points of a diameter are…
- The angle between a tangent to a circle and the radius drawn at the point of…
- From a point Q, the length of the tangent to a circle is 24 cm. and the…
- If AP and AQ are the two tangents a circle with centre O so that ∠POQ = 110°,…
- If tangents PA and PB from a point P to a circle with centre O are inclined to…
- In the figure XY and X^1 Y^1 are two parallel tangents to a circle with centre…
- Two concentric circles of radii 5 cm and 3 cm are drawn. Find the length of the…
- Prove that the parallelogram circumscribing a circle is a rhombus.…
- A triangle ABC is drawn to circumscribe a circle of radius 3 cm. such that the…
- Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct…
- Construct a tangent to a circle of radius 4 cm from a point on the concentric…
- Draw a circle with the help of a bangle, Take a point outside the circle.…
- In a right triangle ABC, a circle with a side AB as diameter is drawn to…
- Draw a tangent to a given circle with center O from a point ‘R’ outside the…
- A chord of a circle of radius 10 cm. subtends a right angle at the centre. Find…
- A chord of a circle of radius 12 cm. subtends an angle of 120o at the centre.…
- A car has two wipers which do not overlap. Each wiper has a blade of length 25…
- Find the area of the shaded region in figure, where ABCD is a square of side 10…
- Find the area of the shaded region in figure, if ABCD is a square of side 7 cm.…
- In figure, OACB is a quadrant of a circle with centre O and radius 3.5 cm. If OD…
- AB and CD are respectively arcs of two concentric circles of radii 21 cm. and 7…
- Calculate the area of the designed region in figure, common between the two…
Exercise 9.1
Question 1.Fill in the blanks
i. A tangent to a circle intersects it in …………….. point (s).
ii. A line intersecting a circle in two points is called a ……………..
iii. A circle can have ……………… parallel tangents at the most.
iv. The common point of a tangent to a circle and the circle is called ……………
v. We can draw ……………….. tangents to a given circle.
Answer:i. One
Property- a tangent to a circle touches it at only one point called the common point.
ii. Secant
Secant- a line that touches the circle at two different points
ii. Two
A circle can have two tangents that are parallel, these two tangents touch the circle on the opposite sides, and distance between the point of contacts is the diameter.
iv. Point of contact
The point where tangent touches the circle is called point of contact.
v. Infinite
We can draw infinite tangents to a circle, as a circle can be assumed to be curve having infinite points and one tangent can be drawn from each point on the circle.
Question 2.A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that P OQ = 12 cm. Find length of PQ.
Answer:![](data:image/jpeg;base64,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We know that tangent to a circle makes a right angle with radius.
∠OPQ = 90°
Applying Pythagoras
PQ2 = OP2 + OQ2
PQ2 = 52 + 122
PQ2 = 25 + 144
PQ2 = 169
PQ = 13cm
Question 3.Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.
Answer:![](data:image/jpeg;base64,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Step1: Draw circle of random radius with BC as diameter.
Step2: Draw line AD perpendicular to BC to touch the circle at D
Such that ∠D = 90°
Step3: Draw line through D parallel to BC that will form a tangent
Step4: Take a random point on line AD as F and draw a line through F parallel to BC to intersect circle in two points that forms a secant.
Question 4.Calculate the length of tangent from a point 15 cm. away from the circle of a circle of radius 9 cm.
Answer:![](data:image/jpeg;base64,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We know that tangent to a circle makes a right angle with radius.
∠OPQ = 90°
Applying Pythagoras
PQ2 = OP2 + OQ2
PQ2 = 92 + 152
PQ2 = 81 + 225
PQ2 = 306
PQ =
cm
Length of tangent =
cm
Question 5.Prove that the tangents to a circle at the end points of a diameter are parallel.
Answer:![](data:image/jpeg;base64,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To prove: DE ∥ FG
Proof:
We know that tangent to a circle makes a right angle with the radius.
Let DE and FG be tangent at B and C respectively.
BC forms the diameter.
∴ ∠OBE = ∠OBD = ∠OCG = ∠OCF = 90°
Also, ∠OBD = ∠OCG and ∠OBE = ∠OCF as alternate angles
∴ DE and FG make 90° to same line BC which is the diameter.
Thus DE ∥ FG
Fill in the blanks
i. A tangent to a circle intersects it in …………….. point (s).
ii. A line intersecting a circle in two points is called a ……………..
iii. A circle can have ……………… parallel tangents at the most.
iv. The common point of a tangent to a circle and the circle is called ……………
v. We can draw ……………….. tangents to a given circle.
Answer:
i. One
Property- a tangent to a circle touches it at only one point called the common point.
ii. Secant
Secant- a line that touches the circle at two different points
ii. Two
A circle can have two tangents that are parallel, these two tangents touch the circle on the opposite sides, and distance between the point of contacts is the diameter.
iv. Point of contact
The point where tangent touches the circle is called point of contact.
v. Infinite
We can draw infinite tangents to a circle, as a circle can be assumed to be curve having infinite points and one tangent can be drawn from each point on the circle.
Question 2.
A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that P OQ = 12 cm. Find length of PQ.
Answer:
We know that tangent to a circle makes a right angle with radius.
∠OPQ = 90°
Applying Pythagoras
PQ2 = OP2 + OQ2
PQ2 = 52 + 122
PQ2 = 25 + 144
PQ2 = 169
PQ = 13cm
Question 3.
Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.
Answer:
Step1: Draw circle of random radius with BC as diameter.
Step2: Draw line AD perpendicular to BC to touch the circle at D
Such that ∠D = 90°
Step3: Draw line through D parallel to BC that will form a tangent
Step4: Take a random point on line AD as F and draw a line through F parallel to BC to intersect circle in two points that forms a secant.
Question 4.
Calculate the length of tangent from a point 15 cm. away from the circle of a circle of radius 9 cm.
Answer:
We know that tangent to a circle makes a right angle with radius.
∠OPQ = 90°
Applying Pythagoras
PQ2 = OP2 + OQ2
PQ2 = 92 + 152
PQ2 = 81 + 225
PQ2 = 306
PQ = cm
Length of tangent = cm
Question 5.
Prove that the tangents to a circle at the end points of a diameter are parallel.
Answer:
To prove: DE ∥ FG
Proof:
We know that tangent to a circle makes a right angle with the radius.
Let DE and FG be tangent at B and C respectively.
BC forms the diameter.
∴ ∠OBE = ∠OBD = ∠OCG = ∠OCF = 90°
Also, ∠OBD = ∠OCG and ∠OBE = ∠OCF as alternate angles
∴ DE and FG make 90° to same line BC which is the diameter.
Thus DE ∥ FG
Exercise 9.2
Question 1.Choose the correct answer and give justification for each.
The angle between a tangent to a circle and the radius drawn at the point of contact is
A. 60o
B. 30o
C. 45o
D. 90o
Answer:Property of tangent- tangent to a circle makes right angle with radius at point of contact
Question 2.Choose the correct answer and give justification for each.
From a point Q, the length of the tangent to a circle is 24 cm. and the distance of Q from the centre is 25 cm. The radius of the circle is
A. 7 cm
B. 12 cm
C. 15 cm
D. 24.5 cm
Answer:Let O be center, OP be radius.
Applying Pythagoras
PQ = 24, OQ = 25
OQ2 = OP2 + PQ2
252 = OP2 + 242
625 = OP2 + 576
OP2 = 49
OP = 7 cm
Radius of Circle = 7 cm
Question 3.Choose the correct answer and give justification for each.
If AP and AQ are the two tangents a circle with centre O so that ∠POQ = 110°, then ∠PAQ is equal to
![](data:image/png;base64,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)
A. 60o
B. 70o
C. 80o
D. 90o
Answer:.
Question 4.Choose the correct answer and give justification for each.
If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80o, then ∠POA is equal to
A. 50o
B. 60o
C. 70o
D. 80o
Answer:![](data:image/jpeg;base64,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We know that tangent to a circle makes right angle with radius.
∴ ∠A = ∠B = 90° and ∠P = 80°
Sum of all angles of quadrilateral = 180°
∴ ∠A + ∠B + ∠P + ∠O = 360°
∴ 90° + 90° + 80° + ∠O = 360°
∠O = 100°
∠ AOP = ∠ BOP
∴ ∠POA = 50°
Question 5.Choose the correct answer and give justification for each.
In the figure XY and X1Y1 are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X1Y1 at B then ∠AOB =
![](data:image/png;base64,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)
A. 80o
B. 100o
C. 90o
D. 60o
Answer:![](data:image/jpeg;base64,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iltbv+3RcbA6+QblNsnfd+HT9a62irUrRcbEuN5J3CiiioKP/Z)
Construct Line OC = radius
OP = OQ = OC = radius
OC∥AP and AC∥OP and
Thus AP = OP = radius
As AP = OP and ∠P = 90°
ΔOAP is isosceles triangle
∴ ∠ PAO = ∠POA = 45°
Also ∠ PAC = 90°
∠OAC = 45° ---1
Also AB is perpendicular to OC
∠OCA = 90°
In ΔAOC
∠OCA + ∠OAC + ∠COA = 180°
45 + 90 + ∠COA = 180°
∠COA = 45°
Similarly
∠BOC = 45°
∴ ∠AOB = 90°
Question 6.Two concentric circles of radii 5 cm and 3 cm are drawn. Find the length of the chord of the larger circle which touches the smaller circle.
Answer:![](data:image/jpeg;base64,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AB = AF = 5cm radius of larger circle
AD = AC = 3cm radius of smaller circle
Pythagoras theorem
AF2 = AD2 + DF2
52 = 32 + DF2
252 = 92 + DF2
DF = 4 units
Length of chord = 2 × DF = 2 × 4 = 8cm
Question 7.Prove that the parallelogram circumscribing a circle is a rhombus.
Answer:![](data:image/jpeg;base64,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FGHI is a parallelogram
∴ HI = FG and FI = GH----1
Also
IB = IE tangents to circle from I
And similarly
HE = HD, CG = GD and CF = BF
Adding all the equations
IE + HE + GC + CF = BF + BI + GD + DH
IH + GF = IF + GH
∴ 2HI = 2HG
HI = HG---2
∴ GF = GH = HI = IF
From 1 and 2
Thus FGHI is a rhombus
Question 8.A triangle ABC is drawn to circumscribe a circle of radius 3 cm. such that the segments BD and DC into which BC is divided by the point of contact D are of length 9 cm. and 3 cm. respectively (See adjacent figure). Find the sides AB and AC.
![](data:image/png;base64,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)
Answer:![](data:image/jpeg;base64,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Construction : Draw radius OB = 3cm
Proof: AC, BC and AB are tangents
BF = BD = 9cm—tangents from B
AF = AB—tangent from A
∴ CD = CB tangents from C
OD = OB = 3cm radius
OBCD forms a square of side 3cm
OD = OB = BC = CD = 3cm
∴ ∠BCD = 90°
BD = 9cm and DC = 3cm
OB = 3cm radius of circle
Let AF = AB = x
Applying Pythagoras
AB = BC + AC
(9 + x)2 = (12)2 + (3 + x)2
81 + 18x + x2 = 144 + 9 + 6x + x2
12x = 72
X = 6cm
∴ AB = 9 + 6 = 15cm
AC = 6 + 3 = 9cm
Question 9.Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct the pair of tangents to the circle and measure their lengths. Verify by using Pythogoras Theorem.
Answer:Step1: Draw circle of radius 6cm with center A, mark point C at 10 cm from center
![](data:image/jpeg;base64,/9j/4AAQSkZJRgABAQEAYABgAAD/2wBDAAoHBwgHBgoICAgLCgoLDhgQDg0NDh0VFhEYIx8lJCIfIiEmKzcvJik0KSEiMEExNDk7Pj4+JS5ESUM8SDc9Pjv/2wBDAQoLCw4NDhwQEBw7KCIoOzs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozv/wAARCADLAPcDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSExBhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD2aiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKTNAC0VTu9WsLFttxdRo/ZM5Y/gOar/ANsSzf8AHnpd3MOzOoiU/wDfXP6UAalFZX2jXZPuWFpCP+mtwWP6Cl/4qA8/8S0e37w0AalFZXma+nJtrCUeiyuufzFL/ad9FzcaPOB3MMiyfpkGgDUorOh17TZZBE1x5EpOPLnUxn9a0AQRkcj1oAWiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigApKR3WNC7sFVRkknAArFNzd66StmzWun5w1zjDzeyeg/2qALV7rMNtP8AZYI3u7s/8sIeSPdj0UfWofsGp6j82o3Ztoj/AMu9o2P++n6n8MVfstPtdPh8q1iEa9SepY+pPUmrNAFSz0uxsB/o1skZPVsZY/UnmrdFFACUtFFABSUtFAEU9tBdRmO4hSVD1V1BFZh0NrQ79Ju5LQ/88mO+I/8AAT0/CtiigDHTWpbSRYdYt/spY4WdDuhY/X+H8a1wwYBgQQeQR3pskaSoySKHRhgqwyDWQ9hdaOxm0oGa26vZM3T3jPY+3SgDaoqtY39vqNuJrdsrnDKRhkPcEdjVmgAooooAKKKKACiiigAooooAKKKKACiiigAooooAKa7rGpd2CqoySTgAUtY1yW1q+axUn7DbN/pLD/lq/aMew6n8qAEVG8RSeZLuXS1P7uPobkj+I/7PoO9bSqFUKoAAGAB2pFUKoCjAAwAO1OoAKKKKACiiigAooooAKKKKACiiigAooooAyr7Tpop21LTcJdAfvI+i3A9D7+hq3YX8OoWyzRZHO10YYZGHVSPWrOM1kajC+m3R1a1QsuMXcSj76/3wP7w/UUAbFFMilSaJJY3Do4DKw6EGn0AFFFFABRRRQAUUUUAFFFFABRRRQAUUUhoAoaveSWtssdtg3Vy3lQA9mPVj7AZNT2FnHYWcdtFkhByx6se5PuTzVGz/ANP1u4vm5itM28H+9/G38h+Fa9ABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUhGaWigDH08nTNRfS2/1EoMtofQfxJ+BOR7Gtis3W7aSWyFxbj/AEm0bzovcjqv4jIq5a3Ed3axXERykqBl+hoAmooooAKKKKACiiigAooooAKKKKACqup3f2HTbi66mKMlR6nt+tWqydc/e/YbPtcXS7h6qvzH+QoAtaTafYdMggbl1XLn1Y8k/mTVykFLQAUUlUJtatILxrWTzgyMqu4iYohb7oLYwM5H50AaFFVY9RtZb+axSTNxCod0weAenP8AnqKs59qAFoqo+pWyXq2hL+YTtyEJUNjIUt0BxzirdABRRSUALRUaSxvI8aurMn3lByV9M+lSUAFFFFABRRRQAUUUUAIaytD/ANGe8009LWbMf+4/zD+o/Ctasp/9H8URMPu3dsyn/eQ5H6MaANWiiigAooooAQ1lrPcTSXExvBbxwT+WEZRtIGM7iecnPGCO3WszxV42h8NXUNotm93PInmEBwgRc46+p54qSy1/w3qkEGptLFHJIAxSQ4ZWHZh0JB71pySjHma0ZHMm+VPVHRCnVl/8JHo4/wCX+M/QE/0o/wCEk0j/AJ/AfpGx/pWZZqUVl/8ACSaV2uHP0hf/AApP+Ej03s85+lu/+FAGrWLqxLaxaKCR5VvPKCPXAA/nUv8AwkVieiXR+ls/+Fef3XxBvl8V/altEa0hka1FuUPmspYAn2fI4GPb3rSnTlNvl6ETnGG/U7TQZZDPGpkV1e1VzsnaQbsjls9Dz/OuhrATxDDHkRaHqg5522mM/rUn/CSOemhasf8At3A/rUyd2OKsjarm7/Q7q51i4nSEATSQulwLll8vZjOU6N0/WrP/AAkVwenh/VD9YlH9aP8AhILw/d8O6ifqEH/s1SURWuhXtte2+o/ai9wZne4iJXy9r9Qp27uMJjJ/hq6dGzq/9obrfO7OPsq7+mPv9ar/ANu6kenhu+/F4x/Wj+29VP8AzLV3+M0f+NACHRp01prqJEAkuBM1wXO4LtCmPHocVu1if2zq56eG7j8biP8Axpf7V1s/d8OsP967QUAbVMdtqlgpbAzgdTWOdR8QHpoEY+t6v+FcB4n8SeK7bxIYy89hIm37NaQneknHHb58nr6VpTpupKyM6k1BXZ3unWd/aX0d1MqN9qDfaAgIKEncpPPOOV49q3BUduZHgjaZQkhQF1HY45FSVDdykrC0UUUigooooAKKKKACsrVv3eo6VMOouTH+DIf8K1ay9b+9px7/AG6PH5GgDTooooAWiiigDC8QeENM8RyxTXnnRzRLtWSF9pK5zg8HitKw0200yyhsrSFUhhXag6n8T3NW6Kpyk1ZvQXKk7jdi/wB0flS4paKkYlLRRQAlc/daRpy+MbG+axgNxJFJ+92DJZduD9QM89a6GsrWP3Vzpt10EdyEY+gcFf54ouBqAcUtJWDaXbGJNUu9UaENM6NA23ywAWGwDruAXOc9jQBv0VznhzWLq8u5obzz1adPtMCywmPapOCgyBux8pzz96r1nq7XOoyWhWABN3KSOW4PoUA/X86ANWiud1G+1G2v9QT7UqxC2jaFVjx5W5ypYk9T39BitDSJJc3dvLNJMLa4MaSSEFmG1W5IHbJFAGlRRRQAlIVBIJAyOhx0p1FACUtFFABRRRQAUUUUAFFFFABWVrHzXmlRjqbvdj2Csa1ayrj994mtIx0t4JJW9ixCj+tAGpRS0UAFFFFABRRRQAUUUUAFFFFABVHWbVrzSbmGP/WFNyf7w5H6ir1IaAILG6W9sYLlOkqBvoe4qNtJ0953neygaWQFXcoMsD1zVTSz9iv7vTG4UN58HujHkD6Nn8616AI2giaRJWjUvHkIxHK564p+KWigCGS0t5i5lhRzInlvuUHcvofbk0W1pb2cIhtoUhjByFQYFTUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAJWXpP8ApOoahf8AVXkEMZ9VTg/+PE1Pq941lpzvGMzPiOFf7ztwP8fwqXT7NbCwgtVOfKQAn1Pc/nmgCzRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAGXrMEqrFqNsu6ezJbaP+WiH7y/lyPcVftriO6t47iFt8cihlPqDUlY0THQ9Q+ztxYXbkwt2hkPVfoe3vQBtUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFJS1l6teTeYmm2LYu7gfe/54p3c/wBPegCOI/2rrJnHNrYEpGezyn7x/wCAjj6k1r1DZWkVjaR20K4jjXA9T6k+9T0AFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFQ3VrDe2z286B43GGBqaigDHs7qXTrhdN1CQsGOLa5bpKP7rHsw/Wteobu0gvbZ7e4jEkbjkH+Y9DWYtzdaJiO+Zriy6JdYy0ftIPT/a/OgDaopqSJIivG6ujDIZTkEU6gAooooAKKKKACiiigAooooAKKSsy61WSSdrLS0W4uRw7k/u4fdj6+w5oAl1LUhZhIII/PvJv9TCO/ufRR60aZp32JHlmk867mO6aYj7x9B6AdhS6dpqWW+V3M91NzLO45b2HoPar1ABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFIQGBBGQeoNLRQBkvpM1k5m0eZYMnLW0nMT/Qfwn6UJr0cLiHVIXsJTwDJzG30ccfnitamyRpKhSRFdT1DDINAAkiSIHjZXU9GU5Bp1ZTeHrVHMljLNYOev2d8Kfqp4pPI123P7u8tbte3nRlG/NeP0oA1qKyft+sx8SaMsnvDcqf5gUf2rf8AfQ7rPtIh/rQBrUVkf2lqr8R6HID6yXCAfpml/wCKgnHAsbQHvlpGH8hQBq1n3WuWNvJ5KObm47QQDex/Lp+NQnQ3uf8AkI6lc3Q7xqfKQ/gv+NaFrZW1lH5drbxwr6IuM0AZxt9S1T/j8f7DbHrBC+ZH/wB5u30H51pW1pBZwLBbRLFGvRVFTUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAJRS0UAJiloooAKKKKACiiigAooooAKKKKACiiigD/9k=)
Step 2: find perpendicular bisector of AC
![](data:image/jpeg;base64,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)
Step3: Take this point as center and draw a circle through A and C
![](data:image/jpeg;base64,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)
Step4:Mark the point where this circle intersects our circle and draw tangents through C
![](data:image/jpeg;base64,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)
Length of tangents = 8cm
AE is perpendicular to CE (tangent and radius relation)
In ΔACE
AC becomes hypotenuse
AC2 = CE2 + AE2
102 = CE2 + 62
CE2 = 100-36
CE2 = 64
CE = 8cm
Question 10.Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also verify the measurement by actual calculation.
Answer:Step1:Draw circles of radius 4 and 6 cm
![](data:image/jpeg;base64,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Step 3: Draw tangent to inner circle from C
![](data:image/jpeg;base64,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AD is perpendicular to DC- tangent and radius
In Δ ADC
AC is radius
AC2 = AD2 + DC2
62 = 42 + DC2
36 = 16 + DC2
DC2 = 20
DC = 2
cm
Question 11.Draw a circle with the help of a bangle, Take a point outside the circle. Construct the pair of tangents from this point to the circle measure them. Write conclusion.
Answer:Step1: Draw a circle with bangle, its center is not known
![](data:image/jpeg;base64,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Step2: to find the center draw 2 random chords C and FE
![](data:image/jpeg;base64,/9j/4AAQSkZJRgABAQEAYABgAAD/2wBDAAoHBwgHBgoICAgLCgoLDhgQDg0NDh0VFhEYIx8lJCIfIiEmKzcvJik0KSEiMEExNDk7Pj4+JS5ESUM8SDc9Pjv/2wBDAQoLCw4NDhwQEBw7KCIoOzs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozv/wAARCAE4AWYDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSExBhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD2aiiigAooooAKKKKACiiigAooooAKKKKACiiigAoqC7vbaxhM11MkKDu5xn6etZv9q39/xpdiRGelxdZRfqF6n9KANjNZ91rumWrGN7pXk/55xZdvyFV10KS6+bVL+a6PeJD5cf5Dr+JrRtrG1s0CW1vHCB/cUCgDP/tbULj/AI8tGmIPR7lhEPy5NHleIp/vXNjag9kjaQj88CtfFLQBkf2PfSc3GuXZ9REqxj9BR/wjsLf6y/1B/rct/SteigDI/wCEbse8l2frdP8A40f8I3Zj7s96v0un/wAa16KAMj+wNo/darqMZ7fv9w/UUn9m6xCP3GtGT2ngVv1GDWxRQBj+b4ht8b7Wzu1HXypDG368UDxBHDxqFld2Xqzx7k/76XNbFJgHjtQBBa31peputbmKYf7DA1Pms+70LTrtvMe3EcvaWI7HH4iq/wBj1mw5s7xb2If8srrhvwcf1FAGzRWTDr8AkEF/E9hOeAs33W+jdDWqDnnt2oAWiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiis/UdWhsWWFUa4upP8AV28f3m9z6D3NAF2WaOGNpZXVEUZZmOAKyP7TvdUJTSIQkPQ3k6/L/wAAXq316UsOkTX0i3OtOszA5S1Q/uo/r/ePua2AoUADgDoBQBm2uh28EouLhnvLr/ntOckf7o6D8K06KKACiiigAooooAKKKKACiiigAooooAKKKKACiiigCOaCK4jMc0aSIequMg1lNpN3p+X0e42oDk2k5LRn/dPVa2aSgDNstainmFrdRtZ3f/PGX+L/AHT0YVpZqve6fa6hAYbqISL1HYqfUHsazPNv9C/15kvrAf8ALUDMsI/2h/EPfrQBuUVFBcQ3MKzQSLJG4yrKcg1IaADIpawo76+W3N88yPCLgxtD5eCF37chs9ehrdptNEqVwooopFBRRRQAUUUUAFFFFABRRRQAUUUUAFFFFACUZqvqFy1pp9xcKu5ooywB+lQW0M0EsbS6g03mL8ySBfmbrlcdB145p20uJvU0KSgdKyL69uL+6bTNMfYy/wDHzc9RCPQerfypDFvdTmmum07SgslyP9bK3KQD39W9qs6dpcOnh2Baa4k5lnk5dz/Qe1S2Njb6farb26bVHJJ5LHuSe5qzQAUUUUAFFFFABRRRQAUUUUAFFFFABRRSUALRSZprSxoPndV+pxQA+iozNGsRmMi+WBkvnjHrmqsOt6ZcXC28N9DJK/3VVsk0AXqKpX2q22nsiziUs4JURxM+cfQVJZX0d/AZo45UUNjEqFD+RoAs0VlDV7xpdiaLd7d2N7FVGM9etW72S8jhBsreOaQtgiSTYAPXpQBZzQap2Lakxc38VvGONghYsffOaguLfWZbhzBqFvBCT8g8jcwHvzQBDcabcadM97pAHzHdNZk4SX1K/wB1v51F/wAJp4fWaO3n1GO3uHIBhlBBjb0bsD9a15YJpbLyRctHKVAMyKM57kA8V5brvw/8QSavdtaKl7FdO0gnllVSCeocfpx2x0rWlGEnabsZ1JSirxVz0ePSFQhTczSQCUyiE7du4nd1AyRn3rTrL8P3EUukW8KM5ktY1hlWQYdWUAHcPwzWpWbbZaSQUUUUhhRRRQAUUUUAFFFFABRRRQAUUUUAFFFFADXVXRlZQysMEEdRVS20y1tpRIiuWVdqb3LeWPRc9BwOnpVw9K87+KOp6hZS6fbx3UttZzK5Zo3Kb3BGASOcY5xWlODnJRT3M6klCLk1sddqd7PLcjStObbcuMyy9rdPX6nsKvWNjBp9qttbphF6k8lj3JPc1xnw5bWZNLaaVI3tZbhj587MZpVwOh7gHIBNdhfQahNsFleR24Gd5aLeT6Y5qZR5ZOJUZcyuW6KrWUFzBEy3V39qcnIbywmB6YFVDojvKXk1W/YbshRKFA56cCpKNWkyM4yM+lV72xhv4PJuN5TcG+Vyp/MVHY6RZaczNaQ+WzjDNuJJ/OgCSbUbG2YrPeQRsOoaQAinyXdvHaG6aVfIC7t45GPXimPp1jJM00lnA8jdXaMEn8anCKqhFUBQMAAcAUAULXXNPvrhbe2meR2BI/dsBx7kU691M2cqxiyu7gsu7MEe4D6mr2O3QUUAV7W5kubbzjbSwMc4jlADVUivdYklQNpCRRlvmZrkEgeuAK1KKAKl8NQKINPNuGz85nzjHtiixTUEV/t80MrE/L5KFQB+J5q3RQBlTabqM0zt/bM0UZbKpHEg2j0zVy5tftVqYGnmjzjLxNtb86s0UAULHSYbGZpUmuZHZdpM0xf9DRc6Lpt5OZ7mzjllIALMM9Kv0UARRW8MNuLeOJFhUbQgHGPTFEdtBFjy4Y0x02qBUtFABRRRQAlGKWigBKKWigBKMUtFAGRqdhPHONU01f8AS0GJI+gnT+6ff0NXbC+h1G1S5gY7W4Kngqe4I9RVqsO+jbRr5tVt1JtpSPtkSjp/00HuO/tQBuUU1HWRA6MGVhkEdCKdQAUUUUAFFFFABRRRQAUUUUAFFFFABRRSZoAZPNHbwSTSsFjjUsxPYCsnS7Q6i8mq38IY3A2wQyLkRxduPU9TSaj/AMTXVY9KXm3hxNdkd/7qfieT9K2gMfSgBAqgBQAAOAB2p1FFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAU11DKVZQwIwQe9OooAxNOZtJ1A6RIx+zyAvZsew7x/h29q2qo6xYHULIrG2y4jIkgf+646f4U7Sr8ajYpPt2ScrKndHHBFAF2iiigAooooAKKKKACiiigAooooAKraheR2FjNdycrEpOPU9h+JqxWNqP/ABMNZtNN6xQ/6TcD1wcIPz5/CgCxolnJa2XmXHN1csZpz/tHt+AwK0qSloAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAQjNYzD+y/EIccW2pfK3oswHH5j9RW1VDWLE6hpssKHbKPnib+645B/OgC8KWqelXo1DTobrGGdfnX0YcEfnVygAooooAKKKKACiiigAooooARiApJOAOprI8PqblLnVHHzXspKe0a8KP5n8ak8QzvDo8yxHEs5EMf1Y4/xq/awJa20VvGMLEgUfgKAJaKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACkpaKAMbTv9B1u+sOkc2LqEfXhx+f862ax9b/ANGu9O1EcCKbypD/ALD8fzxWxQAUUUUAFFFFABRRRQAUUUUAY+pD7Tr2mWnVYy9w4+gwv6mtesm0/f8Aii/l6iCGOEfU5Y/0rXoAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigCjrVsbzRrqAfeaMlf94cj9RUmm3X2zTLa5zkyxKx+uOaskZ4PQ1k+G8pp0lqetrcSRfgGyP0NAGvRRRQAUUUUAFFFFABRRSUAZOg/vJdSuP+el64B9lAH9DWvWP4Y50jzP+ek8r/m5rYoAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAopMiigBaKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigArI0oeXrOrw9B5qSj/gS/wCIrXrHtvk8WXq/89LaNvyJFAGxRRRQAUUUUAFFFFABSE4BNLSUAZXhcf8AFP2x9dx/8eNa1ZHhc/8AEgtx6Fx/48a16ACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooA4fxj44vNC1VNO0+2hd1jWSWSfJHPRQBj0610XhnWv7f0KDUDD5LvlZI85CsDg4PccVBrvhHSfEM8dxexyrNGuwSQyFGK9cH1Famnafa6XYQ2NnEIoIV2ogOcfj3NaydPkSitepnFT5229CzRRRWRoFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFACHgZryKPx/qv/AAkx1Ly4fIdhAbbZz5e/H3uu/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Step 3:Find the perpendicular bisectors of these chords, the points where they intersect is the center of circle
![](data:image/jpeg;base64,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Step 4:
Draw a line from center to a point outside it.
Step 5: draw its perpendicular bisector
![](data:image/jpeg;base64,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Step 6: Cut arcs over the circle with this point as center and point outside it as radius
Step 7: Draw tangent to the circle where the arcs cut the circle
Question 12.In a right triangle ABC, a circle with a side AB as diameter is drawn to intersect the hypotenuse AC in P. Prove that the tangent to the circle at P bisects the side BC.
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)
Answer:ΔABC is right angled triangle
∠ ABC = 90°
Drawn with AB as diameter that intersects AC at P, PQ is the tangent to the circle which intersects BC at Q.
Join BP.
PQ and BQ are tangents from an external point Q.
∴ PQ = BQ ---1 tangent from external point
⇒ ∠PBQ = ∠BPQ = 45° (isosceles triangle)
Given that, AB is the diameter of the circle.
∴ ∠APB = 90° (Angle subtended by diameter)
∠APB + ∠BPC = 180° (Linear pair)
∴ ∠BPC = 180° – ∠APB = 180° – 90° = 90°
Consider ΔBPC,
∠BPC + ∠PBC + ∠PCB = 180° (Angle sum property of a triangle)
∴ ∠PBC + ∠PCB = 180° – ∠BPC = 180° – 90° = 90° ---2
∠BPC = 90°
∴ ∠BPQ + ∠CPQ = 90° ...3
From equations 2 and 3, we get
∠PBC + ∠PCB = ∠BPQ + ∠CPQ
⇒ ∠PCQ = ∠CPQ (Since, ∠BPQ = ∠PBQ)
Consider ΔPQC,
∠PCQ = ∠CPQ
∴ PQ = QC ---4
From equations 1 and 4, we get
BQ = QC
Therefore, tangent at P bisects the side BC.
Question 13.Draw a tangent to a given circle with center O from a point ‘R’ outside the circle. How many tangent can be drawn to the circle from that point?
Hint : The distance of two points to the point of contact is the same.
Answer:Let O be center of circle and R be a point outside it.
Draw tangent to circle at A and B through R
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Choose the correct answer and give justification for each.
The angle between a tangent to a circle and the radius drawn at the point of contact is
A. 60o
B. 30o
C. 45o
D. 90o
Answer:
Property of tangent- tangent to a circle makes right angle with radius at point of contact
Question 2.
Choose the correct answer and give justification for each.
From a point Q, the length of the tangent to a circle is 24 cm. and the distance of Q from the centre is 25 cm. The radius of the circle is
A. 7 cm
B. 12 cm
C. 15 cm
D. 24.5 cm
Answer:
Let O be center, OP be radius.
Applying Pythagoras
PQ = 24, OQ = 25
OQ2 = OP2 + PQ2
252 = OP2 + 242
625 = OP2 + 576
OP2 = 49
OP = 7 cm
Radius of Circle = 7 cm
Question 3.
Choose the correct answer and give justification for each.
If AP and AQ are the two tangents a circle with centre O so that ∠POQ = 110°, then ∠PAQ is equal to
A. 60o
B. 70o
C. 80o
D. 90o
Answer:
.
Question 4.
Choose the correct answer and give justification for each.
If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80o, then ∠POA is equal to
A. 50o
B. 60o
C. 70o
D. 80o
Answer:
We know that tangent to a circle makes right angle with radius.
∴ ∠A = ∠B = 90° and ∠P = 80°
Sum of all angles of quadrilateral = 180°
∴ ∠A + ∠B + ∠P + ∠O = 360°
∴ 90° + 90° + 80° + ∠O = 360°
∠O = 100°
∠ AOP = ∠ BOP
∴ ∠POA = 50°
Question 5.
Choose the correct answer and give justification for each.
In the figure XY and X1Y1 are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X1Y1 at B then ∠AOB =
A. 80o
B. 100o
C. 90o
D. 60o
Answer:
Construct Line OC = radius
OP = OQ = OC = radius
OC∥AP and AC∥OP and
Thus AP = OP = radius
As AP = OP and ∠P = 90°
ΔOAP is isosceles triangle
∴ ∠ PAO = ∠POA = 45°
Also ∠ PAC = 90°
∠OAC = 45° ---1
Also AB is perpendicular to OC
∠OCA = 90°
In ΔAOC
∠OCA + ∠OAC + ∠COA = 180°
45 + 90 + ∠COA = 180°
∠COA = 45°
Similarly
∠BOC = 45°
∴ ∠AOB = 90°
Question 6.
Two concentric circles of radii 5 cm and 3 cm are drawn. Find the length of the chord of the larger circle which touches the smaller circle.
Answer:
AB = AF = 5cm radius of larger circle
AD = AC = 3cm radius of smaller circle
Pythagoras theorem
AF2 = AD2 + DF2
52 = 32 + DF2
252 = 92 + DF2
DF = 4 units
Length of chord = 2 × DF = 2 × 4 = 8cm
Question 7.
Prove that the parallelogram circumscribing a circle is a rhombus.
Answer:
FGHI is a parallelogram
∴ HI = FG and FI = GH----1
Also
IB = IE tangents to circle from I
And similarly
HE = HD, CG = GD and CF = BF
Adding all the equations
IE + HE + GC + CF = BF + BI + GD + DH
IH + GF = IF + GH
∴ 2HI = 2HG
HI = HG---2
∴ GF = GH = HI = IF
From 1 and 2
Thus FGHI is a rhombus
Question 8.
A triangle ABC is drawn to circumscribe a circle of radius 3 cm. such that the segments BD and DC into which BC is divided by the point of contact D are of length 9 cm. and 3 cm. respectively (See adjacent figure). Find the sides AB and AC.
Answer:
Construction : Draw radius OB = 3cm
Proof: AC, BC and AB are tangents
BF = BD = 9cm—tangents from B
AF = AB—tangent from A
∴ CD = CB tangents from C
OD = OB = 3cm radius
OBCD forms a square of side 3cm
OD = OB = BC = CD = 3cm
∴ ∠BCD = 90°
BD = 9cm and DC = 3cm
OB = 3cm radius of circle
Let AF = AB = x
Applying Pythagoras
AB = BC + AC
(9 + x)2 = (12)2 + (3 + x)2
81 + 18x + x2 = 144 + 9 + 6x + x2
12x = 72
X = 6cm
∴ AB = 9 + 6 = 15cm
AC = 6 + 3 = 9cm
Question 9.
Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct the pair of tangents to the circle and measure their lengths. Verify by using Pythogoras Theorem.
Answer:
Step1: Draw circle of radius 6cm with center A, mark point C at 10 cm from center
Step 2: find perpendicular bisector of AC
Step3: Take this point as center and draw a circle through A and C
Step4:Mark the point where this circle intersects our circle and draw tangents through C
Length of tangents = 8cm
AE is perpendicular to CE (tangent and radius relation)
In ΔACE
AC becomes hypotenuse
AC2 = CE2 + AE2
102 = CE2 + 62
CE2 = 100-36
CE2 = 64
CE = 8cm
Question 10.
Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also verify the measurement by actual calculation.
Answer:
Step1:Draw circles of radius 4 and 6 cm
Step 3: Draw tangent to inner circle from C
AD is perpendicular to DC- tangent and radius
In Δ ADC
AC is radius
AC2 = AD2 + DC2
62 = 42 + DC2
36 = 16 + DC2
DC2 = 20
DC = 2 cm
Question 11.
Draw a circle with the help of a bangle, Take a point outside the circle. Construct the pair of tangents from this point to the circle measure them. Write conclusion.
Answer:
Step1: Draw a circle with bangle, its center is not known
Step2: to find the center draw 2 random chords C and FE
Step 3:Find the perpendicular bisectors of these chords, the points where they intersect is the center of circle
Step 4:
Draw a line from center to a point outside it.
Step 5: draw its perpendicular bisector
Step 6: Cut arcs over the circle with this point as center and point outside it as radius
Step 7: Draw tangent to the circle where the arcs cut the circle
Question 12.
In a right triangle ABC, a circle with a side AB as diameter is drawn to intersect the hypotenuse AC in P. Prove that the tangent to the circle at P bisects the side BC.
Answer:
ΔABC is right angled triangle
∠ ABC = 90°
Drawn with AB as diameter that intersects AC at P, PQ is the tangent to the circle which intersects BC at Q.
Join BP.
PQ and BQ are tangents from an external point Q.
∴ PQ = BQ ---1 tangent from external point
⇒ ∠PBQ = ∠BPQ = 45° (isosceles triangle)
Given that, AB is the diameter of the circle.
∴ ∠APB = 90° (Angle subtended by diameter)
∠APB + ∠BPC = 180° (Linear pair)
∴ ∠BPC = 180° – ∠APB = 180° – 90° = 90°
Consider ΔBPC,
∠BPC + ∠PBC + ∠PCB = 180° (Angle sum property of a triangle)
∴ ∠PBC + ∠PCB = 180° – ∠BPC = 180° – 90° = 90° ---2
∠BPC = 90°
∴ ∠BPQ + ∠CPQ = 90° ...3
From equations 2 and 3, we get
∠PBC + ∠PCB = ∠BPQ + ∠CPQ
⇒ ∠PCQ = ∠CPQ (Since, ∠BPQ = ∠PBQ)
Consider ΔPQC,
∠PCQ = ∠CPQ
∴ PQ = QC ---4
From equations 1 and 4, we get
BQ = QC
Therefore, tangent at P bisects the side BC.
Question 13.
Draw a tangent to a given circle with center O from a point ‘R’ outside the circle. How many tangent can be drawn to the circle from that point?
Hint : The distance of two points to the point of contact is the same.
Answer:
Let O be center of circle and R be a point outside it.
Draw tangent to circle at A and B through R
Exercise 9.3
Question 1.A chord of a circle of radius 10 cm. subtends a right angle at the centre. Find the area of the corresponding :
(use π = 3.14)
i. Minor segment
ii. Major segment
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i. Let Major segment A1 minor segment be A2
∠A = 90°
![](data:image/png;base64,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)
![](data:image/png;base64,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)
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![](data:image/png;base64,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)
![](data:image/png;base64,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)
area of sector ACD = 78.5cm2
Pythagoras
AC2 = CD2 + AD2
AC2 = 102 + 102
AC2 = 200
AC = 10![](data:image/png;base64,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)
Height of triangle =
Cos 45° = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
AE = ![](data:image/png;base64,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)
Area of minor segment = Area of sector ACD – Area ΔACD
= 78.5 –
10![](data:image/png;base64,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)
= 78.5-50
= 28.5 cm2
ii. Major segment = Area of circle – minor segment
= Ï€ × r2 –minor segment
= Ï€ × 102 –28.5
= 314-28.5
= 285.5cm2
Question 2.A chord of a circle of radius 12 cm. subtends an angle of 120o at the centre. Find the area of the corresponding minor segment of the circle
(use Ï€ = 3.14 and √3 = 1.732)
Answer:![](data:image/jpeg;base64,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)
Let Major segment A1 minor segment be A2
![](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,iVBORw0KGgoAAAANSUhEUgAAAPcAAAArCAMAAACTru41AAAAAXNSR0IArs4c6QAAAMBQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjoAOjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZjqQZmaQZma2ZpCQZpC2ZpDbZrbbZrb/kDoAkDo6kGY6kJA6kJBmkLaQkLbbkNv/tmYAtmY6tmZmtpA6tpBmttvbttv/tv/btv//25A625Bm27Zm27aQ29u22//b2////7Zm/7aQ/9uQ/9u2/9vb//+2///beH0zWgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAEHklEQVRoQ+1Z61abQBBmiaahrZoaq/ZmpGlrUFsbRAsF2fd/K+fGZvAca44SDaTzw5DszrLf3Gf0vP+0HhK43jdmuB5QNcpy78RL/On6AQfE+evu4746IIz2Csx7U+DGu11Xd7FvejMEGZmxV4z4Oemae5eHhMsr5NPLdy5iwT2A32NzjH/bD/syMEyIB4w53ETg5UghE9y0nuG+DBYz3t9Wisfl3lRHKRv2U4CNuq1I445A9+VoLqe24obI/GbmxYC1IgB+vT+YfwerZjtHyoN2q1mpKeunNtTqtaFRYoCdc9w27E4YjwZguBrO/bjr8mmviePNbXhct16y85rCnb6jmvm3GzcWXtncgUEOHNeUg1e4KQpkHbF0BBX3fp9J6CLYmMeUx8dci1NpaqNu4M4DAJgHrssoPrKic/n07AQT/MYR1mtE3cD9Ik5aTIzRGeRFLvH8L8Xo0UBfIy4JNTWa5MYYi0mkrZPnh7TQG3X9dy/Dw6VCLGVGHvQvvIT7BvDXIlxRVyw5dD5ADOQfVB5+kxCMH+V7DMwkURuq9PTQax65ngRm88d2ak/BXxOOjjl06e/wOFmTk5O3xofgiB0dkEMOXS4FzTkbdvn9PyHuwhggfPoNfOLpMdtNpCY/EZ0r+eeRkBZhi/0xONXAO5vGve9jrPRhADVmsVdrfE7sw2yKVahrfOj70gKr5BobFgoMwfGpN/CB+XZKr9EZV+pO7pWXSHkASY1hxOJU9A0vpNYIK+yUK2ZKQbx9lw9xbNkAjJUCvuKr3sB47Ncp65UaYyGRaF3fkn7BfBobCWJTKj2sqAfVdJMgSLUGt6Jvgls3f1VhWGcDBlGd4qvewBixbCQB0g6hakCygH/L0OExH6ITiSVczkBPY/ytX/hA0xm5AX8j5VWadCrCid0dNsRFylF89fjOIwDY45IZnseiqQ8PnFCaeyCdVFdja3OdnVpzPzOWevMHyRcMtc6GB7HE+Hfiq3f+lP7RxDVu8Y077t28ndOt2O0qn+WLXk75xs4lRQqkDg0gH7Lg6mxadSRZ6oe0d1IE5ZPYzqlr4JOp8l4qwY3T5PNoN/+kDPpVas8ZTbUmMOwp31xHnawHMy6wfhsqNi/ZhnDemxUfZmi5wqeLHVHsJYo8M0eePaejUTQ3P4PhAtXB0+SSBZCSIzNUAxvIx/4XfLFbY/uHpLx1QY86Pv09ACfF7O3YIPv7ePEcCgPY7/j0yMNOjA+ahaoU4eJgdGMHWcjnV7VKXfmZjYVKyf175Gl2oblXfjIZQYVVhI3l+gq7zrfNSbPBk2DeWP17pLlTyxF67erTymtnKSIsJjut0E6z4LEWxPywhnQVLFDid1Eu2YrObJYta26h1om479Nd/Xqgt1iyFAuNALslEJpKL7+3aZ/QbgGcxooS+3CnbgAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
area of sector ACD = 150.72 sq.cm
Sin 30° = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
AE = 6cm
Cos 30° = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
DE = 1.732 × 6
DE = 10.392cm
CD = 2 × 10.392 = 20.784cm
Area of minor segment = Area of sector ACD – Area ΔACD
= 150.72 –
20.784![](data:image/png;base64,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)
= 1550.72-62.352
= 88.44 cm2
Question 3.A car has two wipers which do not overlap. Each wiper has a blade of length 25 cm. sweeping through an angle of 115o. Find the total area cleaned at each sweep of the blades.
(use π = 22/7)
Answer:Radius = 25cm
Angle = 115°
![](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,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)
Area of sector ACD = 627.22cm2
For 2 such wipers: 2 × 627.22
Area = 1254.45 cm2
Question 4.Find the area of the shaded region in figure, where ABCD is a square of side 10 cm. and semicircles are drawn with each side of the square as diameter
(use π = 3.14)
![](data:image/png;base64,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)
Answer:Consider midpoint side AB, it forms Smaller square of size 5cm
For this smaller square
Area of shaded region
= Area of 1st quadrant + area of 2nd quadrant –area of square
Area of both quadrants is same
∴ Area of shaded region
= 2 × Area of quadrant - area of square
= 2 ×
– r × r
= 2 ×
– 5 × 5
= 2 ×
– 5 × 5
= 14.25 cm2
Area of total shaded region = 4 × 14.25
= 57 cm2
Question 5.Find the area of the shaded region in figure, if ABCD is a square of side 7 cm. and APD and BPC are semicircles.
(use π = 22/7)
![](data:image/png;base64,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)
Answer:Area of shaded region = Area of square – area of 2 semicircles
Area of square = 7 × 7 = 49 sq.cm
Area of semicircles = ![](data:image/png;base64,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)
= Ï€ × r2
= Ï€ × 3.52
= 38.5 sq.cm
Area of shaded region = 49-38.5 = 10.5 cm2
Question 6.In figure, OACB is a quadrant of a circle with centre O and radius 3.5 cm. If OD = 2 cm., find the area of the shaded region.
(use π = 22/7)
![](data:image/png;base64,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)
Answer:Area of Shaded region = Area of sector OABC-Area of ΔDOB
OB = OA = 3.5cm
![](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,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)
area of sector OABC = 9.625 cm2
Area of ΔDOB = ![](data:image/png;base64,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)
= 3.5 cm2
Area of Shaded region = 9.625-3.5
= 6.125 cm2
Question 7.AB and CD are respectively arcs of two concentric circles of radii 21 cm. and 7 cm. with centre O (See figure). If ∠AOB = 30°, find the area of the shaded region.
(use π = 22/7)
![](data:image/png;base64,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)
Answer:Area of shaded region = Area(Sector OAB)-Area(Sector OCD)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 102.67 cm2
Question 8.Calculate the area of the designed region in figure, common between the two quadrants of the circles of radius 10 cm. each.
(use π = 3.14)
![](data:image/png;base64,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)
Answer:Area of shaded region
= Area of 1st quadrant + area of 2nd quadrant –area of square
Area of both quadrants is same
∴ Area of shaded region
= 2 × Area of quadrant - area of square
= 2 ×
– r × r
= 2 ×
– 10 × 10
= 2 ×
– 10 × 10
= 57.07 cm2
A chord of a circle of radius 10 cm. subtends a right angle at the centre. Find the area of the corresponding :
(use π = 3.14)
i. Minor segment
ii. Major segment
Answer:
i. Let Major segment A1 minor segment be A2
∠A = 90°
area of sector ACD = 78.5cm2
Pythagoras
AC2 = CD2 + AD2
AC2 = 102 + 102
AC2 = 200
AC = 10
Height of triangle =
Cos 45° =
AE =
Area of minor segment = Area of sector ACD – Area ΔACD
= 78.5 –10
= 78.5-50
= 28.5 cm2
ii. Major segment = Area of circle – minor segment
= Ï€ × r2 –minor segment
= Ï€ × 102 –28.5
= 314-28.5
= 285.5cm2
Question 2.
A chord of a circle of radius 12 cm. subtends an angle of 120o at the centre. Find the area of the corresponding minor segment of the circle
(use Ï€ = 3.14 and √3 = 1.732)
Answer:
Let Major segment A1 minor segment be A2
area of sector ACD = 150.72 sq.cm
Sin 30° =
AE = 6cm
Cos 30° =
DE = 1.732 × 6
DE = 10.392cm
CD = 2 × 10.392 = 20.784cm
Area of minor segment = Area of sector ACD – Area ΔACD
= 150.72 –20.784
= 1550.72-62.352
= 88.44 cm2
Question 3.
A car has two wipers which do not overlap. Each wiper has a blade of length 25 cm. sweeping through an angle of 115o. Find the total area cleaned at each sweep of the blades.
(use π = 22/7)
Answer:
Radius = 25cm
Angle = 115°
Area of sector ACD = 627.22cm2
For 2 such wipers: 2 × 627.22
Area = 1254.45 cm2
Question 4.
Find the area of the shaded region in figure, where ABCD is a square of side 10 cm. and semicircles are drawn with each side of the square as diameter
(use π = 3.14)
Answer:
Consider midpoint side AB, it forms Smaller square of size 5cm
For this smaller square
Area of shaded region
= Area of 1st quadrant + area of 2nd quadrant –area of square
Area of both quadrants is same
∴ Area of shaded region
= 2 × Area of quadrant - area of square
= 2 × – r × r
= 2 × – 5 × 5
= 2 × – 5 × 5
= 14.25 cm2
Area of total shaded region = 4 × 14.25
= 57 cm2
Question 5.
Find the area of the shaded region in figure, if ABCD is a square of side 7 cm. and APD and BPC are semicircles.
(use π = 22/7)
Answer:
Area of shaded region = Area of square – area of 2 semicircles
Area of square = 7 × 7 = 49 sq.cm
Area of semicircles =
= Ï€ × r2
= Ï€ × 3.52
= 38.5 sq.cm
Area of shaded region = 49-38.5 = 10.5 cm2
Question 6.
In figure, OACB is a quadrant of a circle with centre O and radius 3.5 cm. If OD = 2 cm., find the area of the shaded region.
(use π = 22/7)
Answer:
Area of Shaded region = Area of sector OABC-Area of ΔDOB
OB = OA = 3.5cm
area of sector OABC = 9.625 cm2
Area of ΔDOB =
= 3.5 cm2
Area of Shaded region = 9.625-3.5
= 6.125 cm2
Question 7.
AB and CD are respectively arcs of two concentric circles of radii 21 cm. and 7 cm. with centre O (See figure). If ∠AOB = 30°, find the area of the shaded region.
(use π = 22/7)
Answer:
Area of shaded region = Area(Sector OAB)-Area(Sector OCD)
= 102.67 cm2
Question 8.
Calculate the area of the designed region in figure, common between the two quadrants of the circles of radius 10 cm. each.
(use π = 3.14)
Answer:
Area of shaded region
= Area of 1st quadrant + area of 2nd quadrant –area of square
Area of both quadrants is same
∴ Area of shaded region
= 2 × Area of quadrant - area of square
= 2 × – r × r
= 2 × – 10 × 10
= 2 × – 10 × 10
= 57.07 cm2