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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.

Given: ABC is a triangle and P is a point on the side BC such that AB = AP. AP produced meets the circumcircle of ∆ABC at Q.
To Prove: CP = CQ

Proof : $In space increment ABP space and space increment CQP space space space space space space space space space space space angle BAP equals angle QCP$

[ Angles in the same segment of a circle are equal ]

$angle APB equals angle CPQ$
[ Vertically opposite angles ]

$therefore space space space space space space space angle ABP space approximately equal to space angle CQP$

[ AA criterion of similarily ]

$therefore space space space space space space space space space space space space AB over CQ equals BP over QP equals AP over CP$

[ $because$ Corresponding sides of two similar triangles are proportional ]

$rightwards double arrow space space space space space space space space space AB over CQ equals AP over CP$

But AB = AP | Given

$therefore$ CQ = CP

$rightwards double arrow space space space space space space space space space space space space angle straight P space equals space angle straight Q.$

[ Angles opposite to equal sides of a triangle are equal ]

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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.

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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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