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Circles

Long Answer Type

81. In the figure, chord AB of a circle with centre O, is produced to C such that BC = OB. CO is joined and produced to meet the circle in D. If ∠ACD = y and ∠AOD = x, show that x = 3y.

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82. ABC is a triangle and P is a point on the side BC such that AB = A P. If AP produced meets the circumcircle of ∆ABC at Q, prove that CP = CQ.

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Short Answer Type

83. The bisector of ∠B of an isosceles triangle ABC with AB = AC meets the circumcircle of ∆ABC at P as shown in the figure. If AP is produced and meets BC produced at Q, prove that CQ = CA.

Given: The bisector of ∠B of an isosceles triangle ABC with AB = AC meets the circumcircle of ∆ABC at P. AP is produced and meets BC produced at Q.
To Prove: CQ = CA
Proof: ∠ACB = ∠AQC + ∠QAC ...(1)
| An exterior angle of a triangle is equal to the sum of its two interior opposite angles
∵ AB = AC
∴ ∠ABC = ∠ACB    ...(2)
| Angles opposite to equal sides of a triangle are equal
⇒ 2 ∠PBC = ∠AQC + ∠QAC
| ∵ BP bisects ∠ABC and using(1)
But ∠QAC = ∠PBC
| Angles in the same segment of a circle are equal
∴ 2 ∠PBC = ∠AQC + ∠PBC
⇒    ∠PBC = ∠AQC
⇒    ∠PAC = ∠AQC
| ∵ ∠PAC = ∠PBC (Angles in the same segment)
⇒ ∠QAC = ∠AQC
⇒    CQ = CA.
| Sides opposite to equal angles of a triangle are equal

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84. D is a point on the circumference of circumcircle of ∆ABC in which AB = AC such that B and D are on opposite sides of AC. If CD is produced to point E such that CE = BD, prove that AD = AE.

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85. In figure, AB is a diameter of the circle, CD is a chord equal to the radius of the circle. AC and BD when extended intersect at a point E. Prove that ∠AEB = 60°.

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86. Two circles intersect at two points A and B. AD and AC are diameters of the two circles (see figure). Prove that B lies on the line segment DC.

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87. Prove that the quadrilateral formed (if possible) by the internal angle bisectors of any quadrilateral is cyclic.

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88. ABCD is a cyclic quadrilateral in which AC and BD are its diagonals. If ∠DBC = 55° and ∠BAC = 45°, find ∠BCD.

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89. Prove that “The angle subtended by an arc at the centre is double the angled subtended by it at any point on the remaining part of the circle.”

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90. In the figure, O is the centre of the circle. Arc BCD subtends an angle of 140° at the centre. BC is produced to P and CD is joined. Find measure of ∠DCP.

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