Class 9th Mathematics AP Board Solution
Exercise 12.1- Name the following parts from the adjacent figure where ‘O’ is the centre of…
- State true or false. i. A circle divides the plane on which it lies into three…
Exercise 12.2- In the figure, if AB = CD and ∠AOB = 90° find ∠COD w
- In the figure, PQ = RS and ∠ORS = 48°. Find ∠OPQ and ∠ROS.
- In the figure PR and QS are two diameters. Is PQ = RS? infinity
Exercise 12.3- Draw the following triangles and construct circumcircles for them. (i) In ∆…
- Draw two circles passing through A, B where AB = 5.4cm
- If two circles intersect at two points, then prove that their centres lie on…
- If two intersecting chords of a circle make equal angles with diameter passing…
- In the adjacent figure, AB is a chord of circle with centre O. CD is the…
Exercise 12.4- In the figure, ‘O’ is the centre of the circle. ∠AOB = 100° find ∠ADB. (2)…
- In the figure, ∠BAD = 40° then find ∠BCD. 8
- In the figure, O is the centre of the circle and ∠POR = 120°. Find ∠PQR and…
- If a parallelogram is cyclic, then it is a rectangle. Justify.
- In the figure, ‘O’ is the centre of the circle. OM = 3cm and AB = 8cm. Find the…
- In the figure, ‘O’ is the centre of the circle and OM, ON are the…
- A is the centre of the circle and ABCD is a square. If BD = 4cm then find the…
- Draw a circle with any radius and then draw two chords equidistant from the…
- In the given figure ‘O’ is the centre of the circle and AB, CD are equal…
Exercise 12.5
- Name the following parts from the adjacent figure where ‘O’ is the centre of…
- State true or false. i. A circle divides the plane on which it lies into three…
- In the figure, if AB = CD and ∠AOB = 90° find ∠COD w
- In the figure, PQ = RS and ∠ORS = 48°. Find ∠OPQ and ∠ROS.
- In the figure PR and QS are two diameters. Is PQ = RS? infinity
- Draw the following triangles and construct circumcircles for them. (i) In ∆…
- Draw two circles passing through A, B where AB = 5.4cm
- If two circles intersect at two points, then prove that their centres lie on…
- If two intersecting chords of a circle make equal angles with diameter passing…
- In the adjacent figure, AB is a chord of circle with centre O. CD is the…
- In the figure, ‘O’ is the centre of the circle. ∠AOB = 100° find ∠ADB. (2)…
- In the figure, ∠BAD = 40° then find ∠BCD. 8
- In the figure, O is the centre of the circle and ∠POR = 120°. Find ∠PQR and…
- If a parallelogram is cyclic, then it is a rectangle. Justify.
- In the figure, ‘O’ is the centre of the circle. OM = 3cm and AB = 8cm. Find the…
- In the figure, ‘O’ is the centre of the circle and OM, ON are the…
- A is the centre of the circle and ABCD is a square. If BD = 4cm then find the…
- Draw a circle with any radius and then draw two chords equidistant from the…
- In the given figure ‘O’ is the centre of the circle and AB, CD are equal…
Exercise 12.1
Question 1.Name the following parts from the adjacent figure where ‘O’ is the centre of the circle.
i. ![](data:image/png;base64,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)
ii. ![](data:image/png;base64,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)
iii. BC
iv. ![](data:image/png;base64,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)
v. DCB
vi. ACB
vii. ![](data:image/png;base64,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)
viii. shaded region
![](data:image/png;base64,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)
Answer:i. Radius
ii. Diametre
iii. Minor arc
iv. Chord
v. Major arc
vi. Semi-circle
vii. Chord
viii. Minor segment
Question 2.State true or false.
i. A circle divides the plane on which it lies into three parts. ( )
ii. The area enclosed by a chord and the minor arc is minor segment. ( )
iii. The area enclosed by a chord and the major arc is major segment. ( )
iv. A diameter divides the circle into two unequal parts. ( )
v. A sector is the area enclosed by two radii and a chord ( )
vi. The longest of all chords of a circle is called a diameter. ( )
vii. The mid point of any diameter of a circle is the centre. ( )
Answer:i. True
ii. True
iii. True
iv. False
v. False
vi. True
vii. True
Name the following parts from the adjacent figure where ‘O’ is the centre of the circle.
i.
ii.
iii. BC
iv.
v. DCB
vi. ACB
vii.
viii. shaded region
Answer:
i. Radius
ii. Diametre
iii. Minor arc
iv. Chord
v. Major arc
vi. Semi-circle
vii. Chord
viii. Minor segment
Question 2.
State true or false.
i. A circle divides the plane on which it lies into three parts. ( )
ii. The area enclosed by a chord and the minor arc is minor segment. ( )
iii. The area enclosed by a chord and the major arc is major segment. ( )
iv. A diameter divides the circle into two unequal parts. ( )
v. A sector is the area enclosed by two radii and a chord ( )
vi. The longest of all chords of a circle is called a diameter. ( )
vii. The mid point of any diameter of a circle is the centre. ( )
Answer:
i. True
ii. True
iii. True
iv. False
v. False
vi. True
vii. True
Exercise 12.2
Question 1.In the figure, if AB = CD and ∠AOB = 90° find ∠COD
![](data:image/png;base64,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)
Answer:We know “Angles subtended by equal chords at the center of acircle are equal”.
∠COD = ∠AOB = 900
Question 2.In the figure, PQ = RS and ∠ORS = 48°. Find ∠OPQ and ∠ROS.
![](data:image/png;base64,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)
Answer:In
ORS,
∠ORS = ∠OSR (radius of circle)
∠OSR = ∠ORS = 480
∠OSR + ∠ORS + ∠ROS = 180![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
⇒ 48
+ 48
+ ∠ROS = 180![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
⇒ 96
+ ∠ROS = 180![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
⇒ ∠ROS = 180
-96![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
⇒ ∠ROS = 180
-96![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
⇒ ∠ROS = 84![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
We know “Angles subtended by equal chords at the center of a circle are equal”.
∠POQ = ∠ROS = 840
In
POQ,
∠POQ + ∠OPQ + ∠OQP = 180![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
840 + x + x = 180![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
840 + 2x = 180![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
2x = 180
-84![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
2x = 96![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
x = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA4AAAAeCAMAAAArQN94AAAAAXNSR0IArs4c6QAAAHJQTFRFAAAAAAAAAAA6AABmADo6ADqQAGa2OgAAOgA6Ojo6OmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZpDbZrbbZrb/kDoAkLbbkNv/tmYAtmY6tv//25A627Zm29u229vb2////7Zm/9uQ/9u2//+2///bg64kwwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAl0lEQVQoU61Q2xKCIBBdrLSSUEzJChQQ/v8X26UpdabpyTP7ci57hgXgP7xk2ZUipikthKoFX3QQHxeLmts9Ae4CdJ5K/LGHSeVRlc0Jw6AZO0vhCz4O6p0IdY+DzoHYJAVExW0kN6p9mzTGOHVvBrbCZrW/iwZ6/tcK9Q1MhvfOoD9aQIslM/MqynrFHDLXfdKhoqsSfQF3ZQiQhQ8v1QAAAABJRU5ErkJggg==)
x = 48![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
∠OPQ = 48![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
Question 3.In the figure PR and QS are two diameters. Is PQ = RS?
![](data:image/png;base64,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)
Answer:Considering a circle, where PR and QS being a diameter.
Let center of circle be O.
In triangles POQ and SOR
PO = OR (Radius of circle)
OS = OQ (Radius of circle)
Angle POQ = Angle SOR (Vertically opposite angles are always
equal)
Hence triangle POQ is congruent to triangle SOR (By Side Angle Side
Axiom)
PQ = RS (By C.P.C.T.C)
Hence, proved.
In the figure, if AB = CD and ∠AOB = 90° find ∠COD
Answer:
We know “Angles subtended by equal chords at the center of acircle are equal”.
∠COD = ∠AOB = 900
Question 2.
In the figure, PQ = RS and ∠ORS = 48°. Find ∠OPQ and ∠ROS.
Answer:
In ORS,
∠ORS = ∠OSR (radius of circle)
∠OSR = ∠ORS = 480
∠OSR + ∠ORS + ∠ROS = 180
⇒ 48 + 48
+ ∠ROS = 180
⇒ 96 + ∠ROS = 180
⇒ ∠ROS = 180-96
⇒ ∠ROS = 180-96
⇒ ∠ROS = 84
We know “Angles subtended by equal chords at the center of a circle are equal”.
∠POQ = ∠ROS = 840
In POQ,
∠POQ + ∠OPQ + ∠OQP = 180
840 + x + x = 180
840 + 2x = 180
2x = 180-84
2x = 96
x =
x = 48
∠OPQ = 48
Question 3.
In the figure PR and QS are two diameters. Is PQ = RS?
Answer:
Considering a circle, where PR and QS being a diameter.
Let center of circle be O.
In triangles POQ and SOR
PO = OR (Radius of circle)
OS = OQ (Radius of circle)
Angle POQ = Angle SOR (Vertically opposite angles are always
equal)
Hence triangle POQ is congruent to triangle SOR (By Side Angle Side
Axiom)
PQ = RS (By C.P.C.T.C)
Hence, proved.
Exercise 12.3
Question 1.Draw the following triangles and construct circumcircles for them.
(i) In ∆ ABC, AB = 6cm, BC = 7cm and ∠A = 60°
(iii) In ∆ XYZ, XY = 4.8cm, ∠X = 60° and ∠Y = 70°
(iii) In ∆ XYZ, XY = 4.8cm, ∠X = 60° and ∠Y = 70°
Answer:(i) 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)
(ii) 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)
(iii) ![](data:image/png;base64,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)
Question 2.Draw two circles passing through A, B where AB = 5.4cm
Answer:![](data:image/png;base64,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)
Question 3.If two circles intersect at two points, then prove that their centres lie on the perpendicular bisector of the common chord.
Answer:![](data:image/png;base64,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)
Let two circles O and O' intersect at two points A and B so that AB is the common chord of two circles.
OO' is the line segment joining the centers.
Let OO' intersect AB at M
Now Draw line segments OA, OB , O'A and O'B
In ΔOAO' and OBO' , we have
OA = OB (radii of same circle)
O'A = O'B (radii of same circle)
O'O = OO' (common side)
⇒ ΔOAO' ≅ ΔOBO' (SSS congruency)
⇒ ∠AOO' = ∠BOO'
⇒ ∠AOM = ∠BOM ......(i)
Now in ΔAOM and ΔBOM we have
OA = OB (radii of same circle)
∠AOM = ∠BOM (from (i))
OM = OM (common side)
⇒ ΔAOM ≅ ΔBOM (SAS congruency)
⇒ AM = BM and ∠AMO = ∠BMO
But
∠AMO + ∠BMO = 180°
⇒ 2∠AMO = 180°
⇒ ∠AMO = 90°
Thus, AM = BM and ∠AMO = ∠BMO = 90°
Hence OO' is the perpendicular bisector of AB.
Question 4.If two intersecting chords of a circle make equal angles with diameter passing through their point of intersection, prove that the chords are equal.
![](data:image/png;base64,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)
Answer:Given that AB and CD are two chords of a circle, with center O intersecting at a point E.
PQ is a diameter passing through E, such that ∠ AEQ = ∠ DEQ
Draw OL ⊥ AB and OM ⊥ CD.
In right angled
OLE
∠LOE + 90° + ∠ LEO = 180° (Angle sum property of a triangle)
∴∠LOE = 90° – ∠LEO
= 90° – ∠AEQ = 90° – ∠DEQ
= 90° – ∠MEO = ∠MOE
In triangles OLE and OME,
∠LEO = ∠MEO
∠LOE = ∠MOE (Proved)
OE = OE (Common side)
∴ ΔOLE ≅ ΔOME
⇒ OL = OM (CPCT)
Thus, AB = CD
Question 5.In the adjacent figure, AB is a chord of circle with centre O. CD is the diameter perpendicualr to AB. Show that AD = BD.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
Given, AB is a chord of circle with centre O. CD is the diameter
perpendicular to AB.
We know that line drawn from the center of a circle to the chord
Perpendicular to it bisects the chord.
AP = BP
In
ADP and
BDP,
AP = BP
APD = BPD = 90![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
PD = PD (Common)
ADP
BDP
AD = BD (CPCT)
Draw the following triangles and construct circumcircles for them.
(i) In ∆ ABC, AB = 6cm, BC = 7cm and ∠A = 60°
(iii) In ∆ XYZ, XY = 4.8cm, ∠X = 60° and ∠Y = 70°
(iii) In ∆ XYZ, XY = 4.8cm, ∠X = 60° and ∠Y = 70°
Answer:
(i)
(ii)
(iii)
Question 2.
Draw two circles passing through A, B where AB = 5.4cm
Answer:
Question 3.
If two circles intersect at two points, then prove that their centres lie on the perpendicular bisector of the common chord.
Answer:
Let two circles O and O' intersect at two points A and B so that AB is the common chord of two circles.
OO' is the line segment joining the centers.
Let OO' intersect AB at M
Now Draw line segments OA, OB , O'A and O'B
In ΔOAO' and OBO' , we have
OA = OB (radii of same circle)
O'A = O'B (radii of same circle)
O'O = OO' (common side)
⇒ ΔOAO' ≅ ΔOBO' (SSS congruency)
⇒ ∠AOO' = ∠BOO'
⇒ ∠AOM = ∠BOM ......(i)
Now in ΔAOM and ΔBOM we have
OA = OB (radii of same circle)
∠AOM = ∠BOM (from (i))
OM = OM (common side)
⇒ ΔAOM ≅ ΔBOM (SAS congruency)
⇒ AM = BM and ∠AMO = ∠BMO
But
∠AMO + ∠BMO = 180°
⇒ 2∠AMO = 180°
⇒ ∠AMO = 90°
Thus, AM = BM and ∠AMO = ∠BMO = 90°
Hence OO' is the perpendicular bisector of AB.
Question 4.
If two intersecting chords of a circle make equal angles with diameter passing through their point of intersection, prove that the chords are equal.
Answer:
Given that AB and CD are two chords of a circle, with center O intersecting at a point E.
PQ is a diameter passing through E, such that ∠ AEQ = ∠ DEQ
Draw OL ⊥ AB and OM ⊥ CD.
In right angled OLE
∠LOE + 90° + ∠ LEO = 180° (Angle sum property of a triangle)
∴∠LOE = 90° – ∠LEO
= 90° – ∠AEQ = 90° – ∠DEQ
= 90° – ∠MEO = ∠MOE
In triangles OLE and OME,
∠LEO = ∠MEO
∠LOE = ∠MOE (Proved)
OE = OE (Common side)
∴ ΔOLE ≅ ΔOME
⇒ OL = OM (CPCT)
Thus, AB = CD
Question 5.
In the adjacent figure, AB is a chord of circle with centre O. CD is the diameter perpendicualr to AB. Show that AD = BD.
Answer:
Given, AB is a chord of circle with centre O. CD is the diameter
perpendicular to AB.
We know that line drawn from the center of a circle to the chord
Perpendicular to it bisects the chord.
AP = BP
In ADP and
BDP,
AP = BP
APD = BPD = 90
PD = PD (Common)
ADP
BDP
AD = BD (CPCT)
Exercise 12.4
Question 1.In the figure, ‘O’ is the centre of the circle. ∠AOB = 100° find ∠ADB.
![](data:image/png;base64,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)
Answer:Given, ∠AOB = 100°
Join AB, which is a chord.
∠ ACB =
(Angle subtended at the center is twice the angle subtended at circumference by the same chord)
⇒∠ ACB = 50![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
Now, ACBD is a quadrilateral and angles on the opposite sides in the quadrilateral are supplementary.
∠ ADB = 180
-∠ ACB
⇒∠ ADB = 180
-50![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
⇒∠ ADB = 130o
Question 2.In the figure, ∠BAD = 40° then find ∠BCD.
![](data:image/png;base64,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)
Answer:Given, ∠BAD = 40°
Join BD, which is a chord.
We know “Two or more angles subtended by the chord at the circumference are same”.
∠BCD = ∠BAD = 40°
Question 3.In the figure, O is the centre of the circle and ∠POR = 120°. Find ∠PQR and ∠PSR
![](data:image/png;base64,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)
Answer:Given, ∠POR = 120°
We know that “Angle subtended at the center is twice the angle subtended at circumference by the same chord”.
∠PQR = ![](data:image/png;base64,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)
⇒∠PQR = ![](data:image/png;base64,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)
⇒∠PQR = 60![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
Now, PQRS is a quadrilateral and angles on the opposite sides in the quadrilateral are supplementary.
∠ PSR = 180
-∠ PQR
⇒∠ PSR = 180
-60![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
⇒∠ PSR = 120o
Question 4.If a parallelogram is cyclic, then it is a rectangle. Justify.
Answer:Let ABCD be a cyclic parallelogram.
A rectangle is a parallelogram with one angle 90
. So, we have to prove angle 90
.
Since ABCD is a parallelogram,
∠A = ∠C
In cyclic parallelogram ABCD,
∠A + ∠C = 180![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
∠A + ∠A = 180![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
2∠A = 180![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
∠A = ![](data:image/png;base64,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)
∠A = 90![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
Hence, proved.
Question 5.In the figure, ‘O’ is the centre of the circle. OM = 3cm and AB = 8cm. Find the radius of the circle
![](data:image/png;base64,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)
Answer:Joining OA we see that OA is radius of the circle.
Let radius (OA) be r cm.
Given, AB = 8cm
OM = 3cm
We know that “perpendicular from center divides the chord into equal parts”.
AM = ![](data:image/png;base64,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)
⇒ AM = 4 cm
In right angled triangle OAM,
OA2 = OM2 + AM2 (Pythagoras Thm.)
⇒OA2 = 32 + 42
⇒OA2 = 9 + 16
⇒OA2 = 25
⇒OA = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACAAAAAYCAMAAACsjQ8GAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmaQOpDbZgAAZgBmZjoAZjo6ZjqQZrbbZrb/kDoAkDo6kDpmkGY6kNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ29u229v/2////7Zm/9uQ/9u2/9vb//+2///bwMY/3AAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAA3ElEQVQ4T81R2xKCIBSEzOhqV60o6WJC8P8f2AJmmI4zvXVeYNg9u3sOhPxVCdpVn4gmO/bn1ZtHP6EctfFLKMp3hJjbitKhfZXMBhoEBLPPCeF0S1QS4SYZ+I2SM0Tg1kZQYG2CWLz55TfhWVjoVNvxpoU6UKurlzB25cUlmzIf9164jnpIkzovkyFu6qeQDE+iCm3SYB0OATmN8nrPfBTsUyeejdhuSDtijLNEH7e9lQIk4qsfUo5haizIYa8g7YMLOvERuP9wENQa5/xcTabddvtKdPxkf8cP6Av1CQ80MbonMQAAAABJRU5ErkJggg==)
⇒OA = 5 cm
Question 6.In the figure, ‘O’ is the centre of the circle and OM, ON are the perpendiculars from the centre to the chords PQ and RS. If OM = ON and PQ = 6cm. Find RS.
![](data:image/png;base64,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)
Answer:Given, OM = ON
We know that “If two chords are equidistant from the center then the two chords are equal in length”.
PQ = RS = 6 cm
Question 7.A is the centre of the circle and ABCD is a square. If BD = 4cm then find the radius of the circle.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIAAAACHCAIAAABRWcYkAAAAAXNSR0IArs4c6QAAD5tJREFUeF7tnQtUlVUWx9NxpU5rYTMG4yioqb28mCPOlJomTA8xH/hAKFFKAVNERcl0aqXpKtMCszIf+CgwG0FFzJzQmlCb0DJ1LGB0xKmVIFNUI0yrdGDV/PBz3Xu9XO53vtf97uV+d7lYcN1nn3P+/3P22eexz2n1888/X2N9zEOgtXlZWzk3ImARYHI7sAiwCDAZAZOzt3qARYDJCJicvdUDzCagsrKyrq7O5FIEcPat/wv8FgHmtYBWfjcTrjzX+GnduvWdAwaYh5tuOfscARK+J44fu3Tp0tmKipqar/nwi73GoWFhbdr8ov5/9VVVVfYvI/r3b9u2Lf8VGhoWFBTU2xbeq1ev4JAQ3XAyTJGvEHD82LH9+4r2Fb0D7mFhYf0iGgHtCYigGBLCL84IvJiVyZ9zMx6zf0lyEkJeZeU5LGp5WWlFRUVIcMiomJihUVE2W7hhAGpVbDIBB4rf3/PW7gPFxYB+/7DoYdHDXbB2W7+mBLgVKysr3bN798Hi4tq62sioP8ZOiKOjaAVM7/TmEAA0O/Lz39pdSNscNTomMipKkbkQJMCOFT0Dpndsz8eaTYiLHx0zRoRmvaF2r8+rBGAl3sjNeWNLbru27WLj4gDCLe6fHD06K3VGVVVl9PAHunbrdvBAcXh4n/kLFnbu0kWqhFICnJnYnp8H8RRganLKhPh4FSh/8cUXmzZknz51qkfPHh073tC2Xbuvv/rqmWXPqVDVmAQvyAuf2toLKzNfGHjH759e9FTFmTOyOU6e+FDYb3+Tn7cNyerq6lt63mi75aaqykopIar4J6vEg0Bp6Wfz0mdTno3Z6y9evCiu6tDBgxQmbvy477//Xkr16cmTfW29xTW4SBo+E66rq12yeFHU3UMge/eevYuXLFXa/Tt16hQzdixDa1bmCypbWZNkmL6sF1/K31HAPHTQnX+gS9E7ZZV/9+23aanTIWxFZtZ1110nyfe5/fb0efNk0zYnYCwBGJzo++4NDQ0t+egoTou4oW/VqhUlln5ermRffuImqa6n24S4rTSI4kMf8L9Rdw/GB/Osv2Dnzv98912XLqHdu3d3lpyU+LDqghlFAH7hmFEjysvLi959LyllGj6loiJK00P7JFH6pfbCBUVKBIWDgjrQOOid+AWJCROd5xwuGj779CTfBIcEu3zfpk0bwbyaiulPAH0Zm7P06UWLnl66bPkKqqeicC49QPqzw/XXq1AlmITeuWHza1OTk9NmTJcG+aafG4Iboa/5ukZQp4iYzgTg8CUmPITNKdyzV4vT7dIDPj97lsowURCpkhYZpguFb+9FQ3zsOEYvF1Vjx43nG9yz6vPnteTinFZPAg6XlKRMnbLgT09ic/QqH3pw3ncV7Ozatev8xxfoqLY5VVhLLFL6vMfix4/HkDqLhffpM3PW7J9++mllVqbdPNbW1j6/fJnqguk2D8A1ZpCkF6uzOc4V2Pv2nmeWLKGh/a5fxJAhQ44fP3brbbelzZ6D0y2JqZ4HKIKJ3pwxdw7zRJcxtrCgIOf1zfX1DZQKX6h9+/YzZqZ16KDG0lIeHQjA6Gekz8GG4lEoqqFqYe8QQPGk8YxfGMxUl9ZzQq0EsFiJa5wwOZFprUFFbKp2XvqcqspzAwYO8k6OTLU+//xf2wt2sTKoe46axgCWdFKSGo2+N9HXHQJZhbf37cs08OFJCRglWWHFAqrn0MXv/zVm5APnvvxStQbVCbUvRSjNOm782De3vhF97z2sYShN61leZQ+g7a9amZW3o4DJpGLO/TNB9+435m5988mFC/TtB2oIwO5TjtVr1imd3/on8o5S42g8u3wFY57IwpFgZRUTQN6UgFlu4LR9ZyhZxcPpZLYsiK+smGIC8DjxebTMcmXL5OMCbNv1i4hYvuxZXcqpjAAccBaTW7bPIwJratosRgI2dkSEPcsoIIBdvRPHjztvhWvP3n81ZK16aeuWXJe1ChXVESUAwlcsW7Z67VoVebTIJDggbOmw6ItLoqWCQgRIA2/mqlXa13m0lNXX0uKGMAnNmJuuxSkSIuCJhY8nTEr05dM1ZnEzcNCgoZGRWgZkeQIw/WxBqDtAYBYu3syXtXfO8akeDGQIkJYDjVsL9CZSxuXFYMDMVJ0hkiGAzjVpcmJgzrnECcM1vz86es3qV8ST2CU9EUC3onPpu72looh+kYSZwf6iIg8b+s3VwhMBdCuWPvyi/qYXEq80PSNDxWjcLAGsd1Iry/MRp5YlCmZLEm7in2YJWJWVBaXiiixJEAAxcFMEhXsCoBEyoVSRLkuYUy1Ap2hu7J4Aq/mra0yMBEnJKWteXS2e3A0BHEjiJAhkimuxJO0IMGNllVR8TuCGgH3vFBExEWi7XXq1IZbLIiL6s3wgqNANATu258VOUBO5IJhlixcjMI3QKMFquhLAAEJ4G2tMguktsaYI4LwQ9SZohVwJIIKHKCoLVi0IYL2x4bLRBlIWrgRwvtMLh5C1VM8v0rI0xMqESFGvIgD7Q5B0IG+4i0AmIiNuha4iAMtF3xHJwJLxjABWiHZ8uORDWaCuIoCA0KGRFgGyoAkJ3D9s2P59+2RFrR4gC5FKAWayIrMBBwHEt9jCbda2u0q8myRjF6tDUAfZxVEHAWcrzvTsdZNe2Vt6QICdMtktGgcBhCwTXGcBpyMCHOaVXRl1EMDNPEZEgOhYH79TFRwcXFMjE9Pq1APOnbM23/XlGDxlgwkcBDAF49IefUsQ4NqwKNgVzyBYPcDARkKDplkLEcDSnbUBoDsVQCq7JnqlBzBYi19lontBW6pCZV6QD6LAwSS/u1VTKYxXegD3MXA5odLEhsqf/PuJLbk5Rz/+2DkX9quH33cPgTp/fnOr7CTT0OIJKpftBFci5VmHWLUyk7BTQb1eEOPc/fa8bSNGjlqbvcGeHSeFc17b3NDQwDe/7tjx4UemeKEkZME+FbuEXEqqNLvNmzZyE1GPnj2bSyh/PF1plrrI/7u6unPnzu3b//Ld/fv43a6zuvq8hD6fH3/4QZe8RJSwS6gCfTRzz9G1117rKQspjrvkww8JBtc3BlyLtg3r13/zTQ1hOVzdl/XC83ZV69a82qt7N77kH1f6aMnCO2m5F9DzZQJXeoCIwyTSUnSRoY3T0ttytWVsHAq35OTYW/2jM1KTUlJsNtsDI0dmb9qsS3bNKTl96h+DB9zRtXOnBfMz5mfMu7VXj9lpqVyWqChTef9eagiwBFfeaRSyuXAhz+qXX9q8ccNrmzbeEzmUxr5r507ZVEYIjB09ktxxBFA+d/Ysfl+/bq2ijEhiyF0RilqBImHuo+JCDC6mmpKU/MjUpGnTG2PSuetVkRIjhLtdvilRujxNx88VEyTrLemYpWdVmc8v52TGR0eOIHbxxx+5JpJfPv7oyKuvvOy1Mtgz4rJAPg0N9adPn8Ilu/nmW5KnPSpeDJHprePCpptu7FZ66p/WgoQzvuNiRnGP8vjYCfUN9R8cOhQX/+DMtFnX/+pXghywFBoXO447Uz3IO9xQ3+kEgtXzgpjUA7gzjqthps9IzV63dlpykni+IgvMDgJElu7E824Zkvbxk+oMGjyYP48cLhGvGj1AdovFQYDI7oF43i1P8sTlKywVnVoT2WR0IiA0jOcnWh5wqmvEnWzV5xsn4fjESxY/lb1u3YT4B9eud6yLyGoW2ma397L8bdu40V2Rk2sJe0YgecojRe/8RXQe0DvcVl5aJsuqJSCOAOu1smGmDhOEKG+tyG4ii2cf4JKgz8EsBYMweLHiyrp0gAOnV/U5ns4hdVltVy1HExmg+yMJsiVoqQKCJ52vIoAeQHyk7D5yS4VMx3qJR1pcRQAnc3mATvXVNzpWwN9ViUdauO6IcVs7L6v5e/1NLz8YgqRIMVwJGD1mjPSeoEhiS8YtApcjTc8MHHSXCD6uBCiNMxbJI9BkaMFEZwiuK7vZlFcUZxxo4IrUV9z+oM0NAUT4HT5c0vQJG5G8LRmlke5uCKDvcDnx9rw8C00VCHBVSurMNAUJ3S4V8Tole/SKHlm0FuZAAMTADfTE0XB/MIvdMYYRdkEVMGmJcnggNwcDruyYc3NccTSBFzvEmbQkVTR/QGv2aCKLo6zkCd44YbV+EFDT/EnmoeVyWityyF3WSCDSubH7GAxF1l9S6+lwLj2AG9Ckl8ysj2cElj/3LE90K7P+lzXKnI7mREZ5Wam1POcZfU33m8v2L14gp3NZhqg5oEAGQ636PTX5+ADi7WPj4lRcyhsgVovHFVJnzpLdemwODXkCSMn93ewVWx5RUxB5Q5blS02PK8iaIEmAjoYhwhwJygeC2LFPPiGqRaNx9uSGuoBoDQbOgKj2O11QVUAAKTlmxGGjQGjdnutIq6ft0wO0QyE0BthtHwsdjMnqnopoSWMycyN2HBWdE9U0CDsnXvjEkzznFsgDMgMvK/Yu75yrb14qOhEdkJeEdX9ZV0VJvJ+EE7uTJz6kY77KxgB7xsw7zHrNWcfKK1VFm8P019ZeUJrQg7xKAtAorVfrMhDpWB/jVOGA0Ob0RZ/SqieAxJSGFsG5duOq7SOaN2avx/3T6PK7rYsjSE/dMMIJIubizAYXL1mqToOPp5KesmPUNaqCujQxGojuxlGXgmlUwmyLem3JeV2jHkPGABel3DbBkNCSXCOpRkYPcprGABcOJNdod+Eu49qL1zRLfVrFDpfSEiqbCXu21yzJcuMQgQlPLFzgv+e6OFmVmDCR+DrqomKHS/GQppQxEXmMJsdjDDWdIsVQKoOTw4UIlJzZltK0quX17AF28pmmF737Xnl5+ZhRI/xlO5NtxTEjR9TV1RUf+ptXn/BSTZ1IQkYwRgWiX315I4FC4uOzwGBKIfUchJujhJkau6ZUEr9ChDavyTC5ZaSlichG8xpXJK0TMfExhwVU7mBCPmFyIsvagsfnxfWLS+Ig8FgdhcFr4CyJuW92eY8ACSDCYLduySWEatjwaJbUvWptr7mGRwZx0qTcgV42ilqcVNWS3iZAKijz+7cKCwlkIJqZ3sBzK8YxQV4MsITfSk80ErvL4XvVeOme0BwC7NXA6cY08dgNDw7BwdDISH6qPuLhjA5PV4D7wQMHJM2EQNPqffCBFpMJsEMmtVPw4idHYCS73C+iP0NFv4gIblCUJImitU+OSGL3cTHr5WVlOJGc45O+Z+vUzqjuzVZHhb5CgEuVpBsTTlyOGmcH9NKli5IA71zaHwWR3uqSvqdp97bxAlFQb1u48/c6ImWQKh8lwKDa+qBaQ2bCPlhPny2SRYDJ1FgEWASYjIDJ2Vs9wCLAZARMzt7qARYBJiNgcvb/B9UPaU8G82/zAAAAAElFTkSuQmCC)
Answer:Given, BD = 4cm
We know that “a square has equal diagonals” and BD is a diagonal.
AC, which is the radius of the circle, is also a diagonal of the square.
AC = BD = 4 cm
Thus, Radius of the circle = 4 cm
Question 8.Draw a circle with any radius and then draw two chords equidistant from the centre.
Answer:![](data:image/png;base64,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)
Circle with center O and two chords CD and EF equidistant from center .
Question 9.In the given figure ‘O’ is the centre of the circle and AB, CD are equal chords. If ∠AOB = 70°. Find the angles of the ∆OCD.
![](data:image/png;base64,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)
Answer:We know “Angles subtended by equal chords at the center of a circle are equal”.
∠DOC = ∠AOB = 70![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
In
OCD,
∠OCD + ∠ODC + ∠DOC = 180![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
x + x + 70
= 180![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
700 + 2x = 180![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
2x = 180
-70![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
2x = 110![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
x = ![](data:image/png;base64,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)
x = 55![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
∠OCD = ∠ODC = 55![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
In the figure, ‘O’ is the centre of the circle. ∠AOB = 100° find ∠ADB.
Answer:
Given, ∠AOB = 100°
Join AB, which is a chord.
∠ ACB =
(Angle subtended at the center is twice the angle subtended at circumference by the same chord)
⇒∠ ACB = 50
Now, ACBD is a quadrilateral and angles on the opposite sides in the quadrilateral are supplementary.
∠ ADB = 180
-∠ ACB
⇒∠ ADB = 180-50
⇒∠ ADB = 130o
Question 2.
In the figure, ∠BAD = 40° then find ∠BCD.
Answer:
Given, ∠BAD = 40°
Join BD, which is a chord.
We know “Two or more angles subtended by the chord at the circumference are same”.
∠BCD = ∠BAD = 40°
Question 3.
In the figure, O is the centre of the circle and ∠POR = 120°. Find ∠PQR and ∠PSR
Answer:
Given, ∠POR = 120°
We know that “Angle subtended at the center is twice the angle subtended at circumference by the same chord”.
∠PQR =
⇒∠PQR =
⇒∠PQR = 60
Now, PQRS is a quadrilateral and angles on the opposite sides in the quadrilateral are supplementary.
∠ PSR = 180
-∠ PQR
⇒∠ PSR = 180-60
⇒∠ PSR = 120o
Question 4.
If a parallelogram is cyclic, then it is a rectangle. Justify.
Answer:
Let ABCD be a cyclic parallelogram.
A rectangle is a parallelogram with one angle 90. So, we have to prove angle 90
.
Since ABCD is a parallelogram,
∠A = ∠C
In cyclic parallelogram ABCD,
∠A + ∠C = 180
∠A + ∠A = 180
2∠A = 180
∠A =
∠A = 90
Hence, proved.
Question 5.
In the figure, ‘O’ is the centre of the circle. OM = 3cm and AB = 8cm. Find the radius of the circle
Answer:
Joining OA we see that OA is radius of the circle.
Let radius (OA) be r cm.
Given, AB = 8cm
OM = 3cm
We know that “perpendicular from center divides the chord into equal parts”.
AM =
⇒ AM = 4 cm
In right angled triangle OAM,
OA2 = OM2 + AM2 (Pythagoras Thm.)
⇒OA2 = 32 + 42
⇒OA2 = 9 + 16
⇒OA2 = 25
⇒OA =
⇒OA = 5 cm
Question 6.
In the figure, ‘O’ is the centre of the circle and OM, ON are the perpendiculars from the centre to the chords PQ and RS. If OM = ON and PQ = 6cm. Find RS.
Answer:
Given, OM = ON
We know that “If two chords are equidistant from the center then the two chords are equal in length”.
PQ = RS = 6 cm
Question 7.
A is the centre of the circle and ABCD is a square. If BD = 4cm then find the radius of the circle.
Answer:
Given, BD = 4cm
We know that “a square has equal diagonals” and BD is a diagonal.
AC, which is the radius of the circle, is also a diagonal of the square.
AC = BD = 4 cm
Thus, Radius of the circle = 4 cm
Question 8.
Draw a circle with any radius and then draw two chords equidistant from the centre.
Answer:
Circle with center O and two chords CD and EF equidistant from center .
Question 9.
In the given figure ‘O’ is the centre of the circle and AB, CD are equal chords. If ∠AOB = 70°. Find the angles of the ∆OCD.
Answer:
We know “Angles subtended by equal chords at the center of a circle are equal”.
∠DOC = ∠AOB = 70
In OCD,
∠OCD + ∠ODC + ∠DOC = 180
x + x + 70 = 180
700 + 2x = 180
2x = 180-70
2x = 110
x =
x = 55
∠OCD = ∠ODC = 55
Exercise 12.5
Question 1.Find the values of x and y in the figures given below.
i. ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHsAAACACAIAAACk8IHyAAAAAXNSR0IArs4c6QAAEgFJREFUeF7tnQtwVsUVxyu2ExxLmMEJVpsIkiBq0JqgCMgriALKS+VREfAFlZeKgkCxIDhStQWNJTwEfPBSeZqIkYDVgEJ4SRw1iSiICsw4wjhjMggECvSHy9x8uc/de/fefLT5xmE+v+yePfvfs7vnnD179rzTp0//pvYTIQJ1ImyrtqkzCNQiHrUc1CJei3jUCETdXq2M1wTi769f/2VZWdQt/7+2d16tdhjx0Mc14sU7d1ZWVh7Yv//Agf0Cl0+Liysrj5kwatW6jfglNS0tKakhH75EjKN8c/GF+Dd79mwo/HDNO3lgTR8yW7RISEhITklJTk4RXcrIzExIqMuX55+d1qdvv9S0pnzfuqVI/JXqhw4d5MMXKnbM6tShY0f+hYI8ImGXjAvEtxQVrV9XsK5grYCpR89eYO3S89LSkv533dW6TZv5r77mVIzJweBt3LCBf+sn1u+QldWjV6/09OZhA+pJvyYRB2jEGaDT0tJu7dK1S9duksI4ccL41NTUVxbMz1uTzyri2UlGaGNh4Zq8vPKK8r79+vfs1bsGl50aQJx1GbAAOiUlBXEGaBnUDFhZNnr1uL3wo01LFi08cODAU1Of9kTcKEDTK5Yveycvl1+Anv+UmpZvyK0kukpkn5KSL0YOe6h1y+sXzHt5/759/tqdMnkS1al77NixzD9dU17+sw86e3bvfnbaM3ACNd+c+Gj3jGbor5pqraLNmwcNuLtr55vzct9WrRtbHpTTm13Bv+JHA31/NKHD4IH746MfiQz30BE/+OOPQ+6/r99ddxR++IE/XGJrARAoG79AHLyMAfBNf/lbbwl59zdjlNoNEXGAmDXzX/SkYO17Sjw5FYYg1EA5tgDLVMB5Y1BjODu2uwn0tXDrRCQsxJFo1hDWyuACaLAOFkx/U0/YG2hIF0YMJ0306n4bC70umiY6oSD+wvR/smprZxpkwdcKBKuWrmkkiO/85JPgW47TgGn2HVZUlA994H50o0VL39Cr86JNoq3bmjB9+vVbuXy5HtXtVyrYX7nv5q8vKJj61GSNZM+S0jh3xATXK24GexnXNh8x7CHrzsZMYhG4uX17W/EP2DtWdoibdo6ANLVZQMjgnFk5ObPnStqNSrKD0Tjm0Uf79u+/ZPEiLCacAYYjhT/NePElXF2IZM6cuUpkZQrj4Xlywvjp2dm6PAR6EJ+dMxOvHh02sJDpjHwZViqWDrDG4MRoxGci6rJwYbLzhV+y2reVNPrl2xUlaXTog/cPHzkKBlTr2pQPOEdQRdjc0UkC0nGpLqmNmFR1vfzQTfZnYesG/ATSVVhV0UnCVmAlDUuYwejXqIxakYUTq3qqOgD+Eadv7CqY76pNKpVXsiolx0aJAVNhW5tAiaB/xJFuLYa7O7tKICoNjxJMsYUxpIOsoj71cRRVcbyiYSdxJsF+iArE3ohJ0ql9u6aXNxKO1uPHj48cPuyNJUseuHfwRxs2vDhj+pjRj/I7rlc0Co4gQuVqxKiHKyoq8BX7a8UP4q/Mn0djDw79i78m5WvRK+E9R8TWffBhi+tvKHjvPaqD+5bNmwYMHHjy5MnZs3OOHj36008/CbKjx4zJnjFDvgl/JXHKi9MlP9VVJxcGDru2ai0f5U1+qxMnTtzdr8/mTR9D6rFHHs64Jp0vg+8ZkNb4MhPxaJY7Nmp/7hc1GecMBQmakZ3tZ2wV67CeZGa2MM5okOvt27ah+EPm66++MogZurnxyz2DBi1dvFixNeXiiYn1sbxGDR9mZcCDlrz0IXQY8dr9U04MWP1Wq1auSLnkYk8Zp4CTz0u+s5Ilfcx4BRmfOGHcA0OG6vVPOYlDrN/q1KlTH/z7fSBgr27brh1VOnbqdOrUGUz4zpG0lQh8vrpgvrLcqlcQp+FiY5P9SA5mqBadlQfOjAxNv7y8vEPbNrixFi9a+O3evRRmDKZOnoSKMmzokB9++MFa3fbsQrKnqsVoi51D3i6R0sdZSZinoZpzsf2UNOvdoYlSRDgj5fBIEh8pxGMlTlUEfJTXcsIQjTVk9E5+gL3X8RXLlrFUEQAlu04FK0cA2549u4N76VByIOLbTlHtBNbJp8U7RfBeIF0FSWG+RHDCbQiLklnvPoFgHt+Wj0nmrwqLIRq659riIeNY84+NGYvu6TVwev6OJxotZeDge7WQQ8zR6CGohZonERwMGZktvGeVy3iyYSLg/gbcXy08RPiJ/NW1raVlE5bnR2bzcJNxjrgGDhrsOba6CmC8YVgSC6iLIHSQu8T6iYSUaqTpQkpq83AaQJnhkh98mZLy270MNaOMD7NQib6psCdujjI+e1bOg0OGhnRuaRUTBJyA2xEjR2kXRjQW3EEcQGunbEvQ22NsO56mgMogYy5ZN1RJDGn2OHUNbz5Ki9Nf7S0gIvmI55MES0uxUH1PURr9Ag00DieXn/2qQogT0QrRTENacYm30sIDayMqAKuWFmoyRDi3EsdVNh+rhHqu/VqEOpZIBGcIEXcKTwthwLZA2cg4EscBZmR7Jnsahk/YR6ZSepuM9MqVwS/CJUd7o986Dqz6rP3aBdmJoBa/lQy3LnInU121zOKFr/91/DhrLfPOiQuFax+q1H2XjxiFyEYXQJy6Zl5V1q0t6NLN5lRFbjIpl2I3Q+tXrua3QjQn/YI7FhZWZlyhJmbNiG/dWtSqVUSOWb1+K5lRwOhPqJsg5VOVIedVhs3JGmFRkzKO3wYvSmRbtMCHEFmirr2w0vN3zmC5i22mFbsEuxtLvhdrJ7PWeo1KbxNO1FzME70M2Jru1WS8rLTk6qhurhvxVnrESYUK1hDTS6WGz7JM3/Tm6aZFrBriXLpOTk72SV6xGp6yMPxWMlxw4oHNwS4iUzhgGbKPkA4jlsj5U6ZMMf7/7dWrmjW7Mr15WAka6OTnn32GJ+/tVSvLfy6/8qqr+V6/fqJI4BHBBw/lju3bCbggVHPTxx9ztgUD3BbDWgmp9c8//4xGb2h5o0G/2q2U/n3uHP342PAOkYlh27ihkLbBnUijCy+8kO99+vbngk9IHTaRRXMQ2+aJ48e/KPniuusy6tSpw3iTNSSkDZzgIVO6hmqIt7nxhuUrV4dxdUr0HIHis2P7NgJTx44bL34k1UdkCSAQ59KSUtHuwtdeS0pKuq17d7B2T+cSRBrwZ5muhFVDnADtkl1fhzTa8M2Aoy19tWsXEDdo0ED05J5Bg8XtqQg+HL9lvzBdNEQMNJxcl5HB2hLenTG2zaenTM5dk1/VO0MfQpWxRgbr1ZagFvFRrzv/ERj9Vlu/SldhW4tgdhMMjakdgTjLNBGBNcSCYQp3rkKcP4S3noj+M6j4ZsN2zMpgLcqwfFceqwz1CBQhNqmhVYiTQi08JUn0UNdpNYF5BNZ6x+JIYB+lb+ssO8ZKRzwuEZ3aF26DoK5TmLmzZzVLbULoflrjRpOenBicYZwNoSYQMtGvknFUt/D0QoaX293oJMEXrtzVq44c+QWCpJrcsW2rhBx7FMFdHOURqHdsbfAu/YpOJa4MLa7wi/9wicFSQt0LgrMXpdEPt1WII+CIefAO2FLQ6Lea9NSUBhddRCu/r1dv2nPPBWeYacfkYwoGJ2VLwaySGOtgqK7asNfKgKu5rj3Glg1xW8z4VMk4igrqShjjjKOOw5dQN4mAbId30m/VuasQt+rqAbthVI8rq8epU9pTP4mGrHZlFeJWXV0L4rjrRPCjFmrhERGzUHt4v5uM05kwQOf6MJeIw0NKI+UwboHu378vJeWyWCaraYcNkxoerH5gEbA/GNDoP8GvUQVkQ7I6BwMV5RV6jf5DBw8lNUxyRFy7gjgnJ2f4KP0h4ZII+iim3egnZZqRrl7wU03G9a4qbBrFxTvPFQEXcMAtPGs8Aj106BDnHs4ynpzMEZEP0bCtostvpYsfSTqcd8O5ZGHPYlbfSTUZ17iqRB9v5dl5yQJ6jX6PnbNjVtaGwjMnv8E/nO/FpoIMTjAyCtglcO4Yb6/CBwKOdmjO1mEyTLWEMkd/CySglW+qrsvotw1oNvsOO3TMEgEOQT4a/VZB2PBdFw0CTfGd3FzfFERFQhbICWMmYhpeLf6syAL79Ip2LLXgJ+BO9wXNMs7R357de4jr8D28v7420zSaVEO+mfSsGNzoJ26CoEPrCYzNiUTP3r2DTKhzwm/liTgFAmY446kj3t6xNmSDOOUoLcOTtQwhONyDj3+/lUzviDkQwQcyhU1lzubG7N1bCnE2DWLV/J0HkQ8Mf5APFuOzim+jH48piUZss6TYn3O6XQB1xubc8lvJjDFiLi4/yhSOLcOLZjwjZ1/Ldrv3l1mFTNFhZ8YOTzlxouzjTr97Xm57GUfTqJtQV2kJQxBYxNl1VcUhzstj9GN/KilvBDC5xIk4Rk+oLmHnqN/Kc7zPnvQvUzjp98gE5DJP5fNBSOa3X7Tw9QXz5k20u8kb/XLh2aLBrZLR75m2xC3fodM9Zyuvkotd11tuListzby2uWdv46FALLfyb5l5iqlbTFbf/n9Gy/HcqeXT/wwYOKisrPS27j0853I8FIjllpMszrM8uZJKW+IuTTLPJkz/x/NjHxttpXP48OFxYx8nUfjRI0emTP7b1VekxYPkOvGw95tvOmd1yF29GvfTtelXWYvJhPd7CjhkPeIOuROFKeSeSw0D9ZMdO9igTSLAxarhI0aR/X7mS9k3tW0375VXPWWkBgtc3qRJvXr1eCS7ZcuW3MiycnLppZc+8/RUFw65V8Zrzp72tnekJ3nNyYPtlNecedS06RV5+flc6Cem28RQo8aNm6Smfr/v+8633AroNQioTNOEOZw6feo/J0+ynsSWZ13lFuDhXw4n/C7BSWMW7wmTjdO7IZmZ7rIxxs4j23l336CBPW/vJtNKjZeZ8MTYvnfesTY/H/GKDcfE+SySXbukD5NPpiyVm5nGbM+GTE7k2CS3MA1/ZMV/ee6cRn+8pLSk5MuyshrH1J0BEG/b+sbinVXZfEQacCNtr9PZlrxSBwOyiHOLwPqKn1WoDUMfLbD5Vc2KNm/iEl+7Nq2QdPbP+Ed8/BNjYpm0+i2s050zHATcM0GwQVYWcSqYrq3YnpIojXb8DMCbS5fs+rIs+4UZJrGwHmaZbD3+l0mgdKnFe+c0tgK8uJyCPvf3aeIX24BCa0CM904SByXWr1tHFmqSvdS9oNqlC2ucJg5YfCbGpa9Rw4ePnzhRLVBbVdDESuJk+MpvIKrt1kh524sGRt953Jj/VBlTfp9TvOvLUBMtxpV6k4Du3bu3cePGpBeIA8HVwMJ3336Xclny+ef/1kSLM0x8imiEpFBQbUYZcRqgJUDnrEdtNqmyFsflecCAGJNFS9/0c3VPdVKI8pIZ/P0Rj/NatmqbPM8KuoqJKE9Fkv1UvqX/jZK+n3Pzox1aIcPPBejyqui5DjrSjWoQ8NFy/zIu4EPStb9vHp8DIxbSgHAr2JwuKOhiJT6BFlyhEOsSrKAyLhgS0w224hk137xh2WOF6Fo8/WiHtmobevqY0aMJApgw8ck41uvUWBOdQgnm0Vm1mi6lfY+8bUVe80HYo3wjSy//sdRwHLKSaJ+4elaVWEZxeOEAijKDeRigo4aF9PyrfsTpP54HpANXJ1/CgCNUmog2Ki/M61q4TdyGgrhoAzFB2Nl2QgVII3Eg5pkzRBuVVyNZE6kQXU6cSue9m891xW63dI7snS/f+xsn41nt2yYmJua+mx9u7rTwBtOgjMLOdhrGLhSceeSauYhXNrI1MMRVxQQHeylaLetMnMTfolCx4gmslQ5xAg6zNn1ccjrzrsLsWTPFaxXkCA4vK7ETP+L2Arlkt2wp4jSHG8oR5NWMZSZqxEXbWBaAvnLFMl745boq92AigP4M0OsKaJeLwrd27VpTF3xrBnFjzMVlcqK6SJmM1HOOChwan7yFPrewAZoASvZDHnYgwt3PMYLkFJYoVsOIGxwKqedgBYC44ZiRmcmf+Jfk4CkpKZKHTSxZpPoipz1fWD02FhaSLuaMRHfpGuUrXu6wxwvisVwij58WF/ML/5JGMvYaGINhlVBD9cSrQ341HhLgC8VkogAlhFJzkXhE3KWLHOlaIyAj2AM0on6OIa6x5zVFKkSbs6a6FOft1iIe9QDVIh414v8FVYrv6xsKIDMAAAAASUVORK5CYII=)
ii. ![](data:image/png;base64,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)
iii. ![](data:image/png;base64,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)
Answer:(i) We know “Sum of all angles of a triangle is 180
”.
x + y + 30
= 180![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
Since it is an isosceles triangle, x = y.
x + x + 30
= 180![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
⇒ 2x + 30
= 180![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
⇒ 2x = 180
-30![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
⇒ 2x = 150![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
⇒ x = ![](data:image/png;base64,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)
⇒ x = 75![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
x = y = 75![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
(ii) We know that “Angles on the opposite sides in the cyclic quadrilateral are supplementary”.
x + 110
= 180![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
⇒ x = 180
-110![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
⇒ x = 70![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
y + 85
= 180![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
⇒ y = 180
-85![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
⇒ y = 95![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
(iii) Given, x = 90°
x + y + 50
= 180![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
⇒ 90
+ y + 50
= 180![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
⇒ y + 140
= 180![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
⇒ y = 180
-140![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
⇒ y = 40![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAVBAMAAAB4YulwAAAAAXNSR0IArs4c6QAAADBQTFRFAAAAAAAAADpmADqQOgAAOjqQOma2ZgBmkNv/ttv/tv//27Zm2////7Zm/9uQ//+2XNrvcwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAI0lEQVQYV2NgwAV4HTUZGL43bD7A8Izh+wIga+sBBrAYeQAAcb4JQjhC2UwAAAAASUVORK5CYII=)
Question 2.Given that the vertices A, B, C of a quadrilateral ABCD lie on a circle. Also ∠A + ∠C = 180°, then prove that the vertex D also lie on the same circle.
Answer:In a quadrilateral, if the sum of opposite angles is 180°, then it is a
cyclic quadrilateral.
In quad. ABCD, A + C = 180°,
Therefore, ABCD is a cyclic quad.
In a cyclic quad., the vertices lie on the same circle. Thus, D also
lies on the same circle.
Hence, proved.
Question 3.Prove that a cyclic rhombus is a square.
Answer:To prove rhombus inscribed in a circle is a square, we need to prove that either any one of its interior angles is equal to 90° or its diagonals are equal.
![](data:image/png;base64,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)
∠ABD = ∠DBC = b
∠ADB = ∠BDC = a
In the figure, diagonal BD is angular bisector of angle B and angle D.
In triangle ABD and BCD,
AD = BC (sides of rhombus are equal)
AB = CD (sides of rhombus are equal)
BD = BD (common side)
△ABD ≅ △BCD. (SSS congruency)
In the figure,
2a + 2b = 180° (as, opposite angles of a cyclic quadrilateral are always supplementary)
2(a + b) = 180°
a + b = 90°
In △ABD,
Angle A = 180°-(a + b)
= 180°-90°
= 90°
Therefore, proved that one of its interior angle is 90°
Hence, rhombus inscribed in a circle is a square.
Question 4.For each of the following, draw a circle and inscribe the figure given. If a polygon of the given type can’t be inscribed, write not possible.
(a) Rectangle
(b) Trapezium
(c) Obtuse triangle
(d) Non-rectangular parallelogram
(e) Accute issosceles triangle
(f) A quadrilateral PQRS with
as diameter.
Answer:(a) Rectangle = Possible
![](data:image/png;base64,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)
(b) Trapezium = Possible
![](data:image/png;base64,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)
(c) Obtuse triangle = Not Possible
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARkAAADPCAYAAADBAKWRAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAADBwSURBVHhe7Z15kFX1mfefZgdBFlGQRXHHJELAoMQYFCIqBheIwciiiAi+VW8mqZn8MZP3na1qUjNTNeVM5Z2k3qCyCgpB2WSTTQgmGiUCKjvK0qyyb900TTP38+ghLenue+49v3PuOec+v6pT3TT3nvM739/vfM+zPw0uZIbYMAQMAUMgJAQahHReO60hYAgYAoqAkYxtBEPAEAgVASOZUOG1kxsChoCRjO0BQ8AQCBUBI5lQ4bWTGwKGgJGM7QFDwBAIFQEjmVDhtZO7RqCyslLOnTsn/Dx//rxUVFRIVVWV/puD3xn169eXevXq6c8GDRroz4YNG1783fu76/nZ+f4SASMZ2xWxRODs2bNy8uRJOXjwoBw9elSPI0eOSFlZmZSXlwv/D9lw1BTqVVJS8pW/8++mTZtKo0aNlGiaNGkirVu3llatWunPtm3bSsuWLfXgszbcIWAk4w5LO1OeCBw6dEj27dsnBw4ckOPHjwv/5idEgnSC1AEptGjRQo8rrrhCmjVrpsdll112UTqBHPisRxKetONJPKdOnZIzZ84IPyGq0tJS2bx5s/7Odxo3bqzn5PyQT7t27eTKK6+Uq666Spo3b57n3dnXjGRsD0SKAFIHEsr+/ftlx44deiCtQCgQCdJGmzZt5Otf/7o+7PwOsUAmPOioQK4G5ALpHDt2TE6fPi2ff/65Etzhw4dl79698qc//UklJaSbTp06ybXXXivXX3+9Sj7M1YY/BIxk/OFknwqAAA/wrl27ZM+ePSo98JMHHEJp37699O7dWzp27CidO3fWvyGNRDEgCg6IrPqACCEXCIe5MnfmvXHjRiU5SIb5Qjr85LBROwJGMrY7QkEAyYQH8+OPP5YtW7bIiRMnVBVB/ejVq5dKBJ4dJG42EOaD7aZDhw56MF9IEZsQhLNz5079+cknn6jd5+qrr5Zu3brJzTffrPdk46sIGMnYjnCGALYPVI4NGzbIRx99pGrQ5ZdfLjfddJN07dpVH1hsHUkcSDwe6SB5QaLcH0S6fv16WbhwoSxbtkxuuOEGvddbbrlFVTwblrtke8ABArzhMaDyZuehY1x33XXy7W9/W2699Va1qaRtYCRGveO48847Zffu3Xr/HBAs5PqNb3xDCQeSLeZhkkwxr37Ae8dA+u6778ratWtVneBN36dPHzXa4pkploGdBvsMxz333KOesg8//FAJ5w9/+IMSEdIPuKCGFdswkim2FXdwv3iG3n//fX2QCH7r3r279OjRQx+mqIy2Dm4jlFOgIt1444164EXDWAxOv/3tb+Wdd96R73znO0VHNkYyoWy1dJ4UY+cHH3ygxlzIpWfPnqoqFJPUksvKoibecccdcvvttyvZQDKQzcqVK+Wuu+6S2267Tb1paR9GMmlfYQf3hzdl9erVsmnTJnX5fvOb31TxnyA1G9kRQLrz7DNgiAr1xhtvyO9+9zslILxXaTYSG8lk3yNF+wlsLrx1CUrjrdyvXz99+5qbNr8tQToDZINtBrf+e++9J4sWLVLV8/7771e1M43DSCaNqxrwnoj9WLNmjbpkiYi9++671VNE5KuN4AgQh4OLmwOyWbJkibz66quybt06GTBggMYSpWkYyaRpNR3cy9atW1V6YfPjfuUNS0i9jXAQIICvS5cuqkKB+69+9Sv57ne/qwbitKQuGMmEs3cSd1ZC/1etWqXGSVSjwYMHq62g2L1FUSwkbm1c36hRy5cvV8mGWJsHHnhA44ySPoxkkr6CDuZPPAcRqwTV8QblTUowmY1oEcDWNWTIELV7oapOmjRJjeyoUElWVY1kot1HsboaBZ94a+I5IkIX6YWcIhuFRQDphfQEAh2XLl2qeVKsDbE3SRxGMklcNQdzJu9m7ty5sn37djXs3nfffVpPxUY8EECFInoa0ie2Zvz48ao+IWW6LHcRxd0ayUSBcsyuQQTq/PnztVDTE088oSK5jXgigNH9ueeeU1c3a0aO1MMPP5wo9clIJp57K5RZeeoRxl3EcTarBdSFArXTk1Ks6/HHH9e6NRANNW4ee+wxdYEnYRjJJGGVHMwR9Wj27NmqHt17770aWGfqkQNgIzwFsUokoaLmTpw4UdUn1jLuw0gm7ivkYH6I2AR7kSk9fPhw9V7YSCYCZHqPHj1a5s2bJ2+++aYmYeJ9Ipo4riO+M4srYgmbF0F106dP16p0zz77rJWKTNj61TRdkipxdePyXrx4sVYdRJ2Kq2RqJJOCTVfbLZAtPWPGDK2j623KFN9u0d0aKi8vD9RgKvWxxnHsqmAkk9KtiXF3wYIFauD94Q9/mMrqdCldupxui2x4Mrg9N/ewYcO0y0OchpFMnFbD0VyIFkWMpt4LXoi05MA4gid1p8HGhqr02muvaTwNRIOBOC7DSCYuK+FgHhTyJj2AOiWkB3z/+9+PtUHQwS3bKb5EgETLZ555RqZNm6ZEM2LECC0HGofhi2TKy07L8qVLZPP2XdKlc3vp37+/NL+8dRzmb3OohgDJdRDM9773PV2juLUascUKFwHKn2LcnzBhgkyZMkW9UNjjCj2yksyRQwdl7E//TuZtbSL1W98glcfflbsnzZaJ/+9fpPO1ludS6AX0rk/+0VtvvaXxL5RnsFGcCOBxGjlypEozEA3STaGLjGUlmUmvvCYzN7WUNncMl3ol5+WCNJDla2bLf4+bLP/+i38qzpWM2V1TuY6YCertEqBlo7gRoOjVU089JS+//LJMnTpVRo0aVVDDf50kc/5cubz78U5pfN1dUv/CWZHKCimpl2l+fv2dsnbHMqkoPymNmqSvp06StihlGmbNmqUBdo8++mjikueShHWS5kpXS7yKSDMEYj799NMFi6Opk2QglGaN6wlkU1JSLyPFZEbmZ8mFKjl1/KAcO3pcrrraSKZQm++zzz5T1yVJdIMGDZKGDRsWaip23RgiQG4TCbAYgwnI5PdCBOzVSTL16jeUgX17ycS/nyknWoyRJs2aS+XZCinfOE/adD4hr07/rfToebu6SuMYBBTDdXc2JZLkeEPR/J3NQ1CWDUPgUgSQcAljeP3117UQGdJu1A6BrDYZ3pD/WbpX/v/cV+SYtJZ29U9Lhyt3St8+/eTKdu01BZ36pAQF0d7ByCb8jX7q1CmVYNgsTz75ZGL7S4ePlF0BBLDVERFMYiUvJUp9RjmykgzSzE9/+lMZ9qPd8un2TzNFj6+Vz4+ckOkzZsrNmVqwuMlWrFih0aWQTd++fVWyMdE9nGUkFgYj79GjR1XPtlIN4eCctrNSAOvQoUMaR0WgXpT9ubOSjAf2le07CwejXUeRffsPaFQpE8Y373XIwwj5xz/+UTvk0WOmEDpg2jZI9fuByPEm/eAHP7BSmWle6BDuDc9jaWmp5jqNGTMmssJXvknm0numjgWFpxHbsQdQlxRDE2RDzAZ///3vf6+Rp+iFJtkE3zUYeomF+da3vqXtT20YArkgQI4T2dq/+c1v1COJmzuKUp55kwykgREJsR3LNcyI24y2DhAOrlVIZubMmdpmgzqySDZGNrlsiz9/lrohtDYlBoJ0gaiNd/nN2r4VNwTQPKiISJ4TLXCiKHqVN8kAHol3pJe/9NJLGvRDLVJaN8COSC8e2RDqjifkmmuuuahGUSjZhj8E6OiIzevYsWNK5mnum+wPEftUEARw0FDIzDN3kPcU5ghEMkysTZs26kJ98cUX9U07dOjQi3YYqnXR3xcJhtomkA31TWj1QH4NN2dv5OzLi/q5du1adUWSn2LDEAiCAM/cQw89pLWCeWbHjh2rXqewRmCSYWJke6LrIa3g0kaNqj7oQgjZIN3Q75dEPmqUQjKoUVFausMCMqzz0nMHTHn74Iq0YQi4QACHDM8pwgESzY9+9CMXp63xHE5IhjPTVgNxnmrqFM2BPC4dqFE9evRQAzFkg62GRC7UKj5vjcW+ilhVVZUSMm1jyaq2YQi4RACpGI2CZxZtgyOM4YxkmBxBPhiCiePANlNbwWq8UVReR7rBHYtblmQuGrzjjTKy+WKpP/jgA/XWISUmuU1pGBvXzukGAZ5D+m4TP0MVReoHux5OSQZdD88HQT9eGHNdhXMgGyQYpBsajkE2SDZIOnTK69Kli+v7Tcz5jh8/rlIMqqQ1X0vMsiVuojhgeGZRm7CZhlEmxCnJgDCTJlAMjxOxMnhDsjVvx1tSnWxwfXPTiG/8vRiNnbgXT58+rS1MzO2fuGc3UROmDzr2vpUrV+oz57p0p3OSAV08TniZxo0bp0TDg+In8tcjG9So9957T+Ns1q9fr2oVEcS4wIthfPrppxdTNMiwtmEIhI0A8TKbNm3SYE+KXrkcoZAME+ThwGI9efJkNSyRaOnXXY2hkwbw5ECRooDLm+A+/k0iJkF/aR2VlZWyZMkSrWYGsdowBKJAAJsfRmACawmXcKmih0YyAPO1r31N9T0v+5PkyVwGEtGDDz4ovXr10jf7+++/rwDgzuUBLHRZwVzuxe9nuccdO3ZodrUF3flFzT7nAgGIZc2aNfqSwwjMy97FCJVkmCA2lc8//1x98bi2u3XrlvO8+d7AgQM1ZwfJBiMxB/+GgNKSiUzqAHoxLv2w3Ik5g29fKBoECJ5Fg8DMwcuOetEuRugkg4r0yCOPqGsb+wxG4Hy9RlRe51yoTKhQHDAvahSGK/J6kjyQ0qgVg2ctisS1JGNlcw8HAcJHyD+EZEjCdVEfKnSSAQoYkhwnPEakFVCDBlUo34HkAtkgxWAcRrohpgRQ8PuHGSKd75yzfa+srEzvAzE1XxLOdg37f0MgGwIIBWgfSDNoC7zwgo5ISIZJot+R40TQHZnZWLCDJkliAMZdjn0Gmw0HwBDQBwElyaaBYfvw4cMyYMAA3wbyoItv3zcEakIAaYZYNaQZNISgz2lkJMPNdOzYUSUa8pYwBkMQfj1OdW0HyGbw4MEKyNtvv61uOFQpyCcJ9YcpjQhBIsGwuDYMgUIjQCU9Yt1I/+GFHWRESjJMlNQB6llQNAe3mcucHEiMPsB4Z4heJLGQeBskG6KKwwiZDgK+993NmzfL/v371eVPMqkNQ6DQCBCgRwIzFQBIDwrSTz1ykgE8HnpUA1xl2GZwSbscSAQc27dv1yRMcqmQFLDXxI1szp07pwtJVDNeJRuGQBwQwPGAbYY0H3KbgkgzBSEZQMT2gMdpzpw5Gu8SRnNwjKgc27Zt+wuyQY0Kws6uNgIJkNT1QIrBQG7DEIgLAthmkGgwPRBDk296S8F2NRMmuxgrNlX1yHEKK7juxhtvFA4kG9QoiA01ypNs/KQ8hLXwuODxlpkUExbCdt58EUB1R8sg9IT60vlW0CsYyXDjeH+IbMW1TTgzPXvDtJt4kg1V+gh6wy6EC5y8DfKjoraHkK1OUSoM1CbF5Pso2PfCRICo/VatWsmGDRuSSTKAQ4AdPXsnTZp0sXynC49TXcATTQsre50VqOiHSIhFHcN0VA88C8eore5OmJvHzm0I+EGAlz5Ew4uZaOB8gvMKKsl4N8mDTSlAJIt27drpzYQ98P0jvQAgMSoYX1955RUtmIVkwd/DjLrF4EuGOYmk3LMNQyCuCJAKRMwML+V8DMCxIBnAJVWAPk64nXFt53Mz+SwStiGMWoRSExOA+oRkg3cKLxgEGAbZYOzFbU1B57Alt3xwse8YAh4ChIbwIkSawUaT6/MQG5Lhhuhwh8cJiYbUAIy1UQ3IhoRLr7MCZOMVO8dATJCcSzUKKQZRFHKzYQjEGQGeDcwLOE2wI+aakBwrksHwitpEDA3Np/A45XpDQRcLtzZk43VWqK5GkceBZBN0UPGOAkGU1rTavUHRtO9HgQAqE88C+zbXZzJWJANYGJbIcSKk2UumLEQ8C25tEi5RpSh2DsAYpwEbm02QuB4ikunsYG7rKB4Pu4YLBHDQUJkSk0Ku3tDYkQyAoP95OU4QDeU7c9UDXQDLOTAQYy+CXMiSRo1CN/XIJp/6w1u3blUJBpe6DUMgKQhgSpg3b552nyRIz++IJckweVQJOiaiNlG0igp7hRx0ViCexisJirWdcGuPbPzW4sWrRGATkhDntGEIJAUB9iwmjdSQDMDjYcLQtGLFCo0GjkMHRYpu4WJnbtSwIXIYyQYbDmIklvi6BlUCUZWsfm9SHi2bp4cAReE4SktLhf7sfr2isZVkvBujDwwep9mzZ2vkYVxKIXiFl3HpQTaoUp5kQ1BfbcYxpBgWJx81y7a7IVBIBPCu8hLdsmWLlJeX+47Ojz3JeB4n3v6oTs8//3ysgtcgPiQbPFJINZCNV+wc1zcGs+oDURNpKInV+wq5we3a8UAAMwaF4Yjx8muXiT3JAC05Tl7DOIiGHCdXldRdLR1kQ5wPNhvKS5D4yGJQWgLJBrsSxakQNQn0CzNHy9U92XkMgUsRQJJBoiHZOFUkw43icaIcwoQJEzRYj+ZxLoPjXG0ndFYM1nikqGGD+xs1CqkGFYpC4VEGGbq6LzuPIQACSOFUouRl6XckQpLxbgaXL2U2CfunQTgV9uI6UJNoaEesDW5vDlqeMOImhcUVQ5tX/BDgxY49kXgZ+rX7CSZNFMkAOeoIOU6QDCpI3L00iJdkmZOrRIAhbwDqc2Awxo5jtpn4PUg2o7oRIFyDlyae31SSDLdPvAo3SGAQthAypuM+EDEpNQrpIHJS8BwjMWoVZONnseJ+jza/4kAAkiFIlRemn4DSxEkyLCMiGzlOuLZpr/Lcc8/Fvj82pIh46bXYJeaHt8GyZcuUbJDI+FshUiiK49Gwu3SFAKk/7FNyDP2MRJIMN4Z3hhwnqupho3n22WdjLQ1g8K2srLxojyF6koNizUg1CxYsUI8U5SWIIjbvk5/ta58pBAK85FHzMVv4GYklGW4O9QOPE0Tz+uuva45T0EZUfkDL5zPE+VRVVf2F0Rcj2ogRI4R8JlLpCTrEBQ754P7Ot3hzPnO07xgCfhAgmJRnj8BS0mSy7dFEkwyAIA0QQ0P8DMZg1Kg4DkRLcpUwVtc0CHLCtU0qPWSDm57gPtQoUhbiSp5xxNrmFD4CpPlQURITQLYGAIknGeDkjc/N0l+JhxgpIG4D+xG6bF3ua94QlH+gQBAN35BosDnxE7KhXGi2t0bc7tvmk04EeM5Q/zEDFAXJsIz33HOP6ogQDZ6aOBXnPn/+vBw4cEBFTD+DVAo8ZhTIQrKBZFAHPQMx9xZ1ZwU/87bPFA8CPGOUXzl48KBGsNc1UiHJcINIAQMHDlSLNw8kIFBkJw6joqJCGT/XvjUsokc2dDZYvny5TJs2TVVEqvRR36NQdXbigKvNoXAIEIZBYTc/HqbUkAxwY7fwcpwIeKN8ZxyiazH6QjT5Bt5BJBAKNhsiLUlXmDx5smakI8Fhz/Gbdl+4bWlXThMC5BOi/rO3s41UkQw3i0riNYyjqt6wYcMKHnuCvQj91a+6VNuiEZtALA0lQcn0Ro2iVzESEq7vuJTByLbp7P+TjwDqOiRDqgzmgLrU99SRDMuHW5gYGt722GiQbgr5pkdVYhFcVcJDTIVsMARTy4bGdJ5kA9n4icJM/ja3Oyg0ApDM3r171Y1ddCQD+LQawUYzd+5c9Tj17du3YGuCqoRXyHXWOJKNF0+DZAPZEDOEHQdvFGRTSHItGOB24UgQ4GVH+RIkmbpGKiUZ74Z5ACl3+dZbb2nZQOwahRhlZWXK9GG5n9GPkWBw5SPZkK6AGoX6hIGYrpg2DAHXCGDvZG9jCihakuEt/sgjj2iO0/Tp01WHzOZuc70QnM8jGdeSzKVzRR2jQJZXEhQDMZnfqFX8nSRNG4aAKwSQpIliL2pJBjB5sCm1gBpBVDDJlLVF3boC/9LzIFKGoS7VNl8kG7xOlMWAaFCjKHaOpIPEk/be2+crMpJjo6ZhLaed90sEUJcY7O+ilWS8G8enT44Tb3ViaEaOHBlpmL5nfY86pgVxlkLsdFaAaGjjggsc8vEq9aXpiVk4f568sWi1nCg7J91vbC/PPjVU2nXolKZbjNW9eMbeolaXqq+IVzwKLwzG4Cg9ToUiGe/+ic8ZMGDARbKhJCj1h1GrqGeDvSrpY8rkiTLm3+fIuRselYZNmsuMWR/KktU/kzcm/7e0btM26bcXy/l7L01UpqKXZDwA8Lpgo3njjTc0Irh///6RLB4kw4JELclcenPkmOBxw/1NigKlJTiQbCCbXHscRwKej4ucPHZYfj1jlVy4bVRGFW4nF6oqpfmVN8jba17NhDDMlxFPPe3jLPaRXBFAkqH/UtHbZC4FDtcuodB4nHjDU5Wu2AaSCx05IRvUKCQbDv5NTeKobVZB8d9TuksOV14mjVu0kQvnyvR0JfUaSKO2N8qG7XuCnt6+XwsCJsnUsTUefPBBJRrUJh44coHCHHi5YHyOOA1PssFmQ/TwqlWr1G4D8SLZBI1Qjupe27XvIC1KzsiuM6ekSUZCRZKReo2k4sguufZqs8lEtQ61XSfVcTK13TSeHjxO48aN04RDPE7Z0tWDLBRiJXprNt01yDWCfBdvE10gsNEQY7N69WqNtyHGBunGVaRykDnW9d3WbdvJyIe+KX/1q/Fy5Os/lAaNm0n57tXSo9EGGfTo/wrrskV/XtQkXqDZKgIUJcmwO3DzejlOuLZHjx4dWo4TYiUEEzdJ5tKnxCsJihSDGrV06VKVbPBE4f4mziiu48c//t/SpnUrmb54pZwsF+nWu6X8+LlfSLur6+5NHtf7ScK8PK9StvivoiUZFpHeSEg0eJxwbePmzsbK+Sx+UkjGuzc6A3JQXhGpZtGiRVqlD6kmvmRTIsOGj5BhTw6R05lcscta1lyBMJ/1s+/UjIBn8M32zBQ1yQAdhaE8jxMqE61mXQ+YPs7qUm3365ENLUlRoyAbfnopDEiDsRv1G2cI5osgMRvhIuBJMtnSZYqeZFgG1AMMwagHeFZce5zoPEBUJEfc7Rs1bUsSLTkodo5kQ2Y7kcR46pBsknhP4T5+xXH2M2fOqOSfrf60kcyX+wGPEzlOxNDg2nZZLoE3PqxPSnySB8WxOLZt26bFziEb4m3IcCcGKdtmS/K929z/EoHy8nKtjmfqks/dAVB0OkCioY8TVfVcBadhMKXcQ7YcD59TLfjHqNDHsWXLFpX+wItSp17PqEIHHRYcoCKZAHWSSJI0w28OC06uj5fjRPlOGsa56OgI2/uJjMxhqrH4KBX5KCNB8iVqFF461Chc30g2RjaxWKbQJkF1Afa2kUyOEBMzgsdp0qRJ2o5k6NChgR8WJBkkJdSxQpSayBGCnD7OBqMcKKRCHx6C+qZOnXqxOyYtXrKJ0zld0D4cCwTwLFFWloDNbOtrNpkalow3NKoTNWiwzxCCH2TgtcIC76eye5DrFPK72GMwAlMYDMkGL1T1zgqQkI30IHD69GntwIEkm636opFMLetODg9V9VauXKnRwMSI5Dt4ACk3wfnSPiDT6mRDqsKUKVN0M1LjJte2MGnHK6n3RwFxDL+tWrXKegtGMnVAhMcJkZD+1Eg0+T4giJN8vxhIxoPTIxvUJcpK4I2ing8lQSmLap0Vsj6bsf4AUgzDTzqOkUwdS+l5nOgtg1Hz+eefz9vjxGLs27dP2d+FMTnWO7Da5LhX0hK6deummd6oURMnTtRC78TZWP3hpKzkV+fJCxM1ySQZB+tHjAsFrl5++WV11Y4aNSqvhnFIMuix2GUooFVsAxzxOlG7htgavFAbN25UGw6SDW1sbCQHAfYxQaaYAbINk2SyIZT5fzxOQ4YM0TcwqhOJldncdpeeFis8qQV4mIqRZDw8IBuC97DbINmQE4VXCkmHyOu4tBb2sS2K+iPsY6QYr85vXWAYyfjcKgSfDRo0SNUmwH344Yd9fvOLj8H4EBM2HhuiGPbr109TOCgrget7/fr1KtmgRhnZxHeX4L5GkvEbjmEkk8NaUm8FBl+4cKHmOPEw+B0E+iFe8n0bf0YA8oVswBYVCsL56KOPNPYG9YpMeRvxQuDEiRNC3pIfoy8zN5LJcf3uvfdeZXHydngb+43/IImQKnylpaVZewfnOKVUfJyay3jzCBWghg0GYuJt+HeSqvSlYjGy3MSePXs0RcbvC8BIJsddgcpDoN6RI0c0IpgcJ79gI14SN0KMgR+rfI5TS8XHMZDTxgVJxusZBekQt5TE+sOpWJRLbmLXrl0qlfvtcmEkk8cuwC2LIZi4D6JayXHiTZxtUJ9l2bJlKs0YydSNFsmpkDmSDGTjqVJesXPDL9tuC+f/qSSwc+dO7Ubqdw2MZPJcC2wyTzzxhHampKre8OHDs5Y6QIfFBvHpp58WrC93nrdbsK8hJWJwR4qhJCiJmEg2FD/nb343esFuIGUXRgo/dOiQkr/fBFgjmQCbAPWHGBpynKgaR4W9ugbuW9zXZpfJHXRwA2vsM0g1HtlgfMdo7CdeI/er2jcuRWDv3r1atgRvq99hJOMXqVo+R3AZbun58+erx4maKnWNDh06COUssen41WkDTjFVX4dsHn/8cfXs4famfxakw5sV6cbIJtzl3rFjhwajoi75HUYyfpGq43N4nCCNefPmqW2GWI/aBnaZ5cuXCxZ6I5n8wYesKcmBFIMxfcmSJapOkYQJ2fgJEsv/6sX5TYJJd+/erdJ4LqkxRjIO9gs5HATneR4n3qa1BZNhY0BtwniGB8VGMATIfYK4KQnq1R8miphUBaKI8YLYcIMA+xt7DHln2co7VL+ikYwb/NXoi80AjxNV9WgYV5Pozhu2U6dOavwl1sDeuMEXgA1P7WHsBJs3b1Y1ivQPJBsvOTOXN2/wGaXzDEgx1KrONRrbSMbhfiA/yWsYRwzNsGHDaiQRr4oc+q2VPHC3AJANLW44SL70yIbAPiSb2267zUg9ANxgijkgF3sMlzOSCQB6TV8lmxjXNg3jiApGurl00AkB4xlvXSMZxwvw5emoY0P9n02bNinZzJgxQ39iMEaNMgkyN9wp7UDheCTDXLEzkskNa1+fRmelZCeGYDxOGIarD9QoxHuyj++77z7rW+QL1dw/RD0g1gLC4S1M4SximlCjWBP+z2+sR+5XT9c3wI/EyLqcGrXdsZFMSHuB5D7Yf/HixZpIduniYPSl1IEF5oW0ANVOC5FAKBwkX7799tta7BwbDiEHkJCN2hGAXJAIsSX6TaGpfjYjmZB2F/YBwuLJukZURz2iob03MJ5RpwaiydVaH9KUi+K02GVQUdetW6eSDV0pUKuw2eRbXjXtwFHRkZCLhx56KGtngpqwMJIJcYeQTEmOE6kHVNXD44T6xMAbhXSDnQCJx1UjuRBvJzWnBntiaSAc6g+jPk2YMOEi2SDh5OKiTQ0wtdwIhIzqma/90Egm5B2C/cVrGEcL3JEjR2p7FAZeJrohIIoayYS8EDWc3qs/TJW+tWvXankJyAZ7GZINP4t9kEKAg4JYJD9JwCbJFGjHECFJdCoepzlz5qjHiTclrkAO7ARsajNCFmaBIBtyorp37/4VsuHNjYG4uppbmBkW7qp4lKif1L9//7ylO5NkIlo/pJaBAwdqkBhvBBaNwVsU4sEAnEvSWUTTLqrLEB2MixajPBX6kGx+/etfq1qFgZi3eTENWiuDA+kvQaQ6I5kIdw3SCoZg8mwozkR9W2I2CIfHNkOIvEkzES5ILZeCbPAOsjZ0ViBNYcOGDWpD69Onj3pZimF89tlnGhszYMCAnHKVLsXGSCbi3UKJScTPuXPnqh0GLxPkgzRDPlOxvS0jhj+ny3kSJ0ZijMPUsSFeBOmTNUu7HQ1yxSsaNMfOSCanbRf8wxh9KVWAx4lYjbFjx2rxJdypSDRGMsExdn0GCmPxcvDIhrADOitANqxdrmH2rucXxvnIU0J6o30NRBNkGMkEQS/P7zZv3vxijhOubYgGA+OsWbMEEdWIJk9gQ/4a4QdEcmMk5i2PvQIXOG966tmkiWy4P6oFQKxBh5FMUATz/D6Rk0g0eJwIdaeqHnVRsM0YyeQJakRfg2wITINYkD6RbCAbCpiRG5X0OkEE3yGpYX8KKsWwJEYyEW3Mmi5DODvkggRD9C+eDcp4Up6zWIyLBYQ/8KUhG6K6kWwwECPZEG8D+eCNcvGABp5kHifgRUfAIqTpYhjJuEAxwDkgFgzBeJxockbwHvYZAvgs6jQAsBF+lRcERctYS4IrkW6QbCAbKvflG8QW4S1cvBRSDBG+GLYpXeJiGMm4QDHgOXAR0pWPNwiBe0QAb9261XJpAuIa9ddJhCXQEmMwa0n7G2wbSDr8DRtHnAdxMStWrFB3NfN1NYxkXCEZ4DzkhRCoR7Ie1eD5N5nCdENAbLWRLASoKYQkimTjkQ3ub8gGFQTDfxwHpUeQYpDKiONyNYxkXCEZ8DyoSaQeTJkyRY4dO6ZeJjZmtu4HAS9rXw8RAdIROKiASPQwZT/orADZ4LWhdXFcRnl5uarszBc1z+UwknGJZsBz4XHCkPjaa68p0WCboZSkl7kd8PT29QIhgETKUZ1seIEg6SDZxKHYOXYkbIOjR4++mMDrCi4jGVdIOjoPNU2w0VCMnJ7DqE01lfB0dDk7TYQIeGRDnhptcYj69uoPQza5lrV0NXUC73ihYYdhjq6HkYxrRB2cD1GatwrlO9mEFLVCorGRDgTIUUMtIaKW9aUECJINcSkk0kZph6OXEmoSBEfPqjCGkUwYqDo4J1naJFMuWLBAC5KzMaPcfA5uwU5RBwIY98nu5gUC2SBJEP2N0Rg7HP9H0bOwB4GEJEEiLYflag//LsJGKaXnZxMSqEfVPJLzaMeKB8pGuhAg657sbgIzqSuENwpVmTUnE5y/sxfCGEjLS5cu1RIj5GGFNYxkwkLWwXmJq6C9yv79+1Wk5i1HYSUb6UMAIiEHCnUJyQayIeUECRYDMX/3Kiq6uHsq3rGn+EmKRJhSk5GMixUL8RxEk1Ky84UXXtDSkD//+c9TX2IgRDhjf2pUYsgGNYr8Ibw+qFGUBEGyQY1yMciTI+CTeB76ioc5jGTCRNfRufE4jRo1Sn75y1/KxIkT5Sc/+UnBPBGObslOkwUBpBZSEqqXBH3llVdUfYJsaBCY70BSIrIX24+r/KS65mIkk+9KRfw93IsE67HRKOGJGmUj/QigxlBBEQmGJEyC+ei3jvoE2eTqcj5y5IjuH6SXBx54IBIAjWQigdnNRYifIaALXZraJbg8bRQHAriYIRUkDzxCkM24cePUaIxE4qfYOfYXYnMqKytl0KBBgUpq5oK6kUwuaBX4s+jrqE2HDh3S9AMMwVp75sI5kZIv2qzYSDcCOAMgG+w2a9asUS8U6g82HMgG281XR9WX/6ynGeJ8Ngo7TPU5GMkkbE9So4Qmcf/1ny/IP/zfv5Py+m1k/4kK6XxFU/nZ80OlZ69vJ+yObLr5IMA+oJoidhvIhmxvEhyRbOiv3qhBPRk/cYrMXble6mVUrq5XN5MWTetnvtMvEjuMkUw+qxqj7/C2ur1Hdxn+s5XS4I4fS4PW18jq3Vtk+cj/I4un/Jt07+kuTT9Gt21TqQEBj2yoyOdJNi+/NE7eefd9mb/rKml4y6CMlFtflq2YLfe22yn/+I//HDmOJslEDrmDC1ZVyqI/bJTGvX8iLa+5LaMtnZHLb+4tBzOBXf/yX+PlX/++lVyQEgcXslMkBQHibLDT9evXV6ZNmSgLtzaQVt99RuqXZNSlTJ2Yc3c9Ix9tnJYxHr8nfTLSTJTDSCZKtB1d63xFmazffkAatL9fCUYuVGV+npYWHW6Vd9fMkOnTJkmlNMrsrQuOrminSQICVFJs0fwy2Vu6W0ra3CoNGzWRqvLjmalfyATyNZbPz7eRLZnYGCOZJKxmgedYv1FT6dqppXxyYIdcdn3PDMGUZao1N5Xy0vdlwO23yuNPDJOqCyVGMgVep0JcvmHG/nJ9l86y6IWlUl52Rho3+KLoWUXGo3RZxR65rkv0NjuTZAqxE4Jes14D+ZsxT8iKsb+Qw1IljVt2lLNH10vJ+hdl0H/8tdzS9WtBr2DfTzAC117bWXq8NEWWrB4nTbsPysgxIuWbFsuzdzWTe/pGqyoBo5FMQjfTHb2/I4te+gf5j3HTZcvu30vPrh2l24AR8snGLVo1nwAuG8WHwMmTJ2X69Olyz129pF/fZvLWewul4lyFDBrZU54bOTSToxR9qIORTIL3Yc9evWVarzulKmOjqdfoi1KOZPBSWe/MmTMWrJfgtc1n6sRPzZgxQw4cOCBDhw6VW265Rf72/Fk1/EqDJvmc0sl3jGScwFjIk5RcJBhmQV0QQtGJ7CwrK5P777/fWqsUcnkiujYF6Hm5nDp1SoYPHy433XTTF1eu3ziiGdR+GSOZgi+B2wlQn+Sxxx7TBEpKPB4/flxDyF2WCXA7YztbUATIpp45c6a+TJ566qmc85mCXj/b941ksiGUwP9ns1EjhEAtJBoq0SPhxL3vTwKhLviUSRPABkMjNlSkOLbINZIp+DYJbwLkuFAJn4TKqVOnypAhQ6RVq1bhXdDOHCkCJEqSUU1DQNbWZa8klzdiJOMSzRieCy8TRMPb7sUXX9TN6CdjN4a3YlP6EgGKf6MKU5K1W7duMnjw4Fj1cLp0oYxkimDrkqFLdb1Zs2bJ+PHjtVYw5GO9tpO3+NR8prg8yZBkXVP+I+4F5o1kkrfP8poxtWLHjBmjbVZwc9L7h3akcepimNeNFdGX6JHOi+Ls2bMqkSYlFspIpog2KYbgJ598UuvQLFq0SOMpIBqtSWMjtgicO3dOy2XS6I+KdiNGjJBOnTrFdr6mLiVmacKZKCoSBmE2K0ZDSjkict99993hXNDOGggB2pbMmTNHNm/erOoR/bji0NY2l5sySSYXtFL0WQpRjx07VtUn3Nz79u3TYkdx9VCkCHrft4L3aPHixaoeUd85KeqRSTK+lzj9H2zevLmWYkR9olUpQV20Ku3du3doDcXSj2rwO9yzZ4+qRxh3abyGpBl225Lgs679DCbJhIluAs6N+oSqRBg63QQRzen3QzpCkLYbCbj12E3x9OnT2q6W/tikhuAFTAPhG8nEbqsVZkI0kSNilHalGIVxdSOeU0fWVKjw1wTPEarRwYMH5c4771T7yxVXXBH+hSO4gpFMBCAn5RJINfTzoYYwb1O6FyKyY2yEcMLqyZwUfMKYJ4mNeI3og035TOKZLiY3hnHBApzTSKYAoMf9kthqUJdoKIZUQ1oCrTdQq+hgaLE1wVcQcqFZ24cffqjJrOSa0cCP39M2jGTStqIO74c369NPP32xJzO1aq666iptAI9a1aRJ4WqUOLzNSE+FF4/GbGvXrtVIXZq10WkgjomNroAxknGFZErPQ+kIGonRz2fLli369n3zzTdVssF2wP+ZZJN98en8ScVCjOpIK9hc6JnUtm3b7F9O+CeMZBK+gFFNH28H9hrUpW3btsk777yjZIPdBg8I/1cMD0wueNMWdvv27bJu3Trt3OiRC+RcTNnwRjK57Br7rCZVYpjkgGwwEBNjs2rVKnV5d+3aVW6++eairV1z/vx5IYkRUvn4449l//79WuuF+CNUTH4vtmEkU2wr7vB+CRTjIHiMtzWeKNSByy+/XNUrCIe8KKSgtI+jR4/Kxo0blVxKS0u1HQ1JqdivyIIvZpUy/auf9t0dg/ujaBIHb+vqhIPdpn379uqlgox4i6flYYNEjhw5opIKRlxsLoT/Yyzv06ePqo/cuw1riWJ7wCEClPdEVeI4ceKEqlPEfxAiz4EdgoewS5cuehDklxSXLaRCRC6uZ8pk7N69W9UiirW3bNlS44iQ3MiOtniir24qk2QcPmR2qj8jgMqEe5aDkhK86Xft2qWqBGoVth1IBrIh+I/cHNy4cVKtIBXmzryZPwRDqxnmCFlyb6hE5H4lLTM6yr1qJBMl2kV6LVIWOPCq8OanfAFqFZIObvE1a9bo2x+JgAOyIaQe9Yp/UwcHiSeMCnCoODREgzxQf5BOmN+xY8e00wNEw9zwnGFnglRQgyBIk1j8bWgjGX842accIcAbH5WCA9LhISZADSmBB5wHG+Lh73hqeJCx4/A9SAfCQS3jbwQDctDuhc8hYRDXw+C71MLl4HcKP3FO+hLhWuYn14JgOCAbpCuIjGtAbkhYkCNSFpKLjfwQMJLJDzf7liMEIAzPS8UpsX1AAth0ePiRKlBZ+B1JAwmoMtM8HuKg1QvE4BELv3t1izkPh3dOfveICFLiQKWDPJBQkJ6QTkipYE4QltVAdrPIRjJucLSzOEKABxvViIeeA/XEG5APEgdSCQe/Qzj8nZ8cSC4MCAXyqU4snsoFgSCxIB0ZkThauDpOYyQTPsZ2BUcIQAxh2GUcTc9OUwsCRjK2NQwBQyBUBIxkQoXXTm4IGAJGMrYHDAFDIFQEjGRChddObggYAkYytgcMAUMgVASMZEKF105uCBgCRjK2BwwBQyBUBIxkQoXXTm4IGAJGMrYHDAFDIFQEjGRChddObggYAkYytgcMAUMgVASMZEKF105uCBgCRjK2BwwBQyBUBIxkQoXXTm4IGAL/A5+Ixl3bkv1xAAAAAElFTkSuQmCC)
(d) Non-rectangular parallelogram = Not Possible
![](data:image/png;base64,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)
(e) Acute isosceles triangle = Possible
![](data:image/png;base64,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)
(f) A quadrilateral PQRS with
as diameter = Possible
![](data:image/png;base64,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)
Find the values of x and y in the figures given below.
i.
ii.
iii.
Answer:
(i) We know “Sum of all angles of a triangle is 180”.
x + y + 30
= 180
Since it is an isosceles triangle, x = y.
x + x + 30
= 180
⇒ 2x + 30 = 180
⇒ 2x = 180-30
⇒ 2x = 150
⇒ x =
⇒ x = 75
x = y = 75
(ii) We know that “Angles on the opposite sides in the cyclic quadrilateral are supplementary”.
x + 110
= 180
⇒ x = 180-110
⇒ x = 70
y + 85
= 180
⇒ y = 180-85
⇒ y = 95
(iii) Given, x = 90°
x + y + 50
= 180
⇒ 90 + y + 50
= 180
⇒ y + 140 = 180
⇒ y = 180-140
⇒ y = 40
Question 2.
Given that the vertices A, B, C of a quadrilateral ABCD lie on a circle. Also ∠A + ∠C = 180°, then prove that the vertex D also lie on the same circle.
Answer:
In a quadrilateral, if the sum of opposite angles is 180°, then it is a
cyclic quadrilateral.
In quad. ABCD, A + C = 180°,
Therefore, ABCD is a cyclic quad.
In a cyclic quad., the vertices lie on the same circle. Thus, D also
lies on the same circle.
Hence, proved.
Question 3.
Prove that a cyclic rhombus is a square.
Answer:
To prove rhombus inscribed in a circle is a square, we need to prove that either any one of its interior angles is equal to 90° or its diagonals are equal.
∠ABD = ∠DBC = b
∠ADB = ∠BDC = a
In the figure, diagonal BD is angular bisector of angle B and angle D.
In triangle ABD and BCD,
AD = BC (sides of rhombus are equal)
AB = CD (sides of rhombus are equal)
BD = BD (common side)
△ABD ≅ △BCD. (SSS congruency)
In the figure,
2a + 2b = 180° (as, opposite angles of a cyclic quadrilateral are always supplementary)
2(a + b) = 180°
a + b = 90°
In △ABD,
Angle A = 180°-(a + b)
= 180°-90°
= 90°
Therefore, proved that one of its interior angle is 90°
Hence, rhombus inscribed in a circle is a square.
Question 4.
For each of the following, draw a circle and inscribe the figure given. If a polygon of the given type can’t be inscribed, write not possible.
(a) Rectangle
(b) Trapezium
(c) Obtuse triangle
(d) Non-rectangular parallelogram
(e) Accute issosceles triangle
(f) A quadrilateral PQRS with as diameter.
Answer:
(a) Rectangle = Possible
(b) Trapezium = Possible
(c) Obtuse triangle = Not Possible
(d) Non-rectangular parallelogram = Not Possible
(e) Acute isosceles triangle = Possible
(f) A quadrilateral PQRS with as diameter = Possible