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Circles

Short Answer Type

71. ABC is an isosceles triangle with AB = AC. A circle through B and C intersects AD and AC at D and E respectively. Prove that BC || DE.

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72. Prove that the circle drawn on any equal side of an isosceles triangle as diameter, bisects the third side.

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73.

In the figure, D is the centre of a circle. Prove that
2(∠XZY + ∠YXZ) = ∠XPZ]

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Long Answer Type

74. Prove that the opposite angles of an isosceles trapezium are supplementary.

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75. In the figure, O is the centre of the circle. BD = OD and CD ⊥ AB. Find ∠CAB.

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76. If the bisectors of the opposite angles of a cyclic quadrilateral ABCD intersect the circle circumscribing it at the points P and Q, prove that PQ is a diameter of the circle.

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Short Answer Type

77. If O is the centre of a circle as shown in figure, then prove x + y = z.

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78.

5. In the given figure, find the values of a, b, c and d. Given that ∠BCD = 43° and ∠BAE = 62°.

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79. P and Q are centres of the two circles which intersect at B and C. ACD is a straight line. Find the values of x, y, z.

2x = 150°
| The angle subtended by an arc of a circle at the centre is twice the angle subtended by it at any point on the remaining part of the circle
⇒    x = 75°    ...(1)
x + ∠BCD = 180°
| ∵ ACD is a straight line
⇒ 75° + ∠BCD = 180°
⇒    ∠BCD = 105°
∵ BEDC is a cyclic quadrilateral
∴ ∠BCD + ∠BED = 180°
| Opposite angles of a cyclic quadrilateral are supplementary
⇒ 105° + y = 180°
⇒ y = 75°    ...(2)
z = 2y
| The angle subtended by an arc of a circle at the centre is twice the angle subtended by it at any point on the remaining part of the circle
⇒    z = 2 × 75° = 150°    ...(3)

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80. ABCD is a cylic quadrilateral. If AC bisects both the angles A and C then prove that ∠ABC = 90°.

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