Question

Bisectors of opposite angles of a cyclic quadrilateral PORS
intersect the circle, circumscribing it at the points A and B. If the
radius of the circle is 12 cm, then length of AB is​

Answer 1

Answer:

Given:ABCD is a cyclic quadrilateral.

DP and QB are the bisectors of ∠D and ∠B, respectively.

To prove: PQ is the diameter of a circle.

Proof: Since,ABCD is a cyclic quadrilateral.

∴∠CDA+∠CBA=180∘ since sum of opposite angles of cyclic quadrilateral is 180∘

21∠CDA+21∠CBA=21×180∘=90∘

⇒∠1+∠2=90∘         ........(1)

But ∠2=∠3 (angles in the same segment QC are equal)  ........(2)  

⇒∠1+∠3=90∘

From eqns(1) and (2),

∠PDQ=90∘

Hence,PQ is a diameter of a circle, because diameter of the circle subtends a right angle at the circumference of the circle.

answer by swati