In the adjoining figure, ABCD is a cyclic quadrilateral. The line PQ is the tangent to the circle at A. If ∠CAQ : ∠CAP = 1 : 2, AB bisects ∠CAQ and AD bisects ∠CAP, then find the measures of the angles of the cyclic quadrilateral. Also prove that BD is a diameter of the circle.
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Question

In the adjoining figure, ABCD is a cyclic quadrilateral. The line PQ is the tangent to the circle at A. If ∠CAQ : ∠CAP = 1 : 2, AB bisects ∠CAQ and AD bisects ∠CAP, then find the measures of the angles of the cyclic quadrilateral. Also prove that BD is a diameter of the circle.

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KnowledgeBoat · Accepted Answer

Given, AB and AD are bisectors of ∠CAQ and ∠CAP respectively. Let AB bisects ∠CAQ in two halves of each value x. ∴ ∠CAB = x and ∠BAQ = x. Let AD bisects ∠CAP in two halves of each value y. ∴ ∠CAD = y and ∠DAP = y. Since, ∠CAQ and ∠CAP are linear pair so, ⇒ ∠CAQ + ∠CAP = 180° ⇒ ∠CAB + ∠BAQ + ∠CAD + ∠DAP = 180° ⇒ x + x + y + y = 180° ⇒ 2x + 2y = 180° ⇒ x + y = ⇒ x + y = 90°. ∴ ∠CAB + ∠CAD = 90° ⇒ ∠BAD = 90°. Since angle in semicircle is equal to 90°. Hence, proved BD is the diameter of the circle. Given, ∠CAQ : ∠CAP = 1 : 2. Let ∠CAQ = k so ∠CAP = 2k. Since ∠CAQ and ∠CAP are linear pair so, ⇒ ∠CAQ + ∠CAP = 180° ⇒ k + 2k = 180° ⇒ 3k = 180° ⇒ k = 60°. ∠CAQ = 60° and ∠CAP = 2 × 60° = 120°. From figure, ∠ADC = ∠CAQ = 60° (∵ angles in alternate segments are equal) ∠ABC = ∠CAP = 120° (∵ angles in alternate segments are equal) Since sum of opposite angles in cyclic quadrilateral is 180°. ⇒ ∠D + ∠B = 180° ⇒ 60° + ∠B = 180° ⇒ ∠B = 180° - 60° ⇒ ∠B = 120°. Similarly, ⇒ ∠A + ∠C = 180° ⇒ 90° + ∠C = 180° ⇒ ∠C = 180° - 90° ⇒ ∠C = 90°. Hence, the angles of cyclic quadrilateral are ∠A = 90°, ∠B = 120°, ∠C = 90° and ∠D = 60°.

Mathematics

In the adjoining figure, ABCD is a cyclic quadrilateral. The line PQ is the tangent to the circle at A. If ∠CAQ : ∠CAP = 1 : 2, AB bisects ∠CAQ and AD bisects ∠CAP, then find the measures of the angles of the cyclic quadrilateral. Also prove that BD is a diameter of the circle.

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Related Questions

In a triangle ABC, the incircle (centre O) touches BC, CA and AB at P, Q and R respectively. Calculate :
(i) ∠QOR
(ii) ∠QPR, given that ∠A = 60°.

In the figure (i) given below, AB is a diameter. The tangent at C meets AB produced at Q, ∠CAB = 34°. Find :
(i) ∠CBA
(ii) ∠CQA

In the figure (i) given below, O is the centre of the circle. The tangents at B and D meet at P. If AB is parallel to CD and ∠ABC = 55°, find
(i) ∠BOD
(ii) ∠BPD.

In the figure (ii) given below, O is the centre of the circle. AB is a diameter, TPT' is a tangent to the circle at P. If ∠BPT' = 30°, calculate
(i) ∠APT
(ii) ∠BOP

Mathematics

In the adjoining figure, ABCD is a cyclic quadrilateral. The line PQ is the tangent to the circle at A. If ∠CAQ : ∠CAP = 1 : 2, AB bisects ∠CAQ and AD bisects ∠CAP, then find the measures of the angles of the cyclic quadrilateral. Also prove that BD is a diameter of the circle.

Circles

Related Questions

In a triangle ABC, the incircle (centre O) touches BC, CA and AB at P, Q and R respectively. Calculate :(i) ∠QOR(ii) ∠QPR, given that ∠A = 60°.

In the figure (i) given below, AB is a diameter. The tangent at C meets AB produced at Q, ∠CAB = 34°. Find :(i) ∠CBA(ii) ∠CQA

In the figure (i) given below, O is the centre of the circle. The tangents at B and D meet at P. If AB is parallel to CD and ∠ABC = 55°, find(i) ∠BOD(ii) ∠BPD.

In the figure (ii) given below, O is the centre of the circle. AB is a diameter, TPT' is a tangent to the circle at P. If ∠BPT' = 30°, calculate(i) ∠APT(ii) ∠BOP

In a triangle ABC, the incircle (centre O) touches BC, CA and AB at P, Q and R respectively. Calculate :
(i) ∠QOR
(ii) ∠QPR, given that ∠A = 60°.

In the figure (i) given below, AB is a diameter. The tangent at C meets AB produced at Q, ∠CAB = 34°. Find :
(i) ∠CBA
(ii) ∠CQA

In the figure (i) given below, O is the centre of the circle. The tangents at B and D meet at P. If AB is parallel to CD and ∠ABC = 55°, find
(i) ∠BOD
(ii) ∠BPD.

In the figure (ii) given below, O is the centre of the circle. AB is a diameter, TPT' is a tangent to the circle at P. If ∠BPT' = 30°, calculate
(i) ∠APT
(ii) ∠BOP