Question

the bisector of opposite angles <P and <R of a cyclic quadrilateral PQRS intersect the corresponding circle at A and B respectively..

prove that AB is a daimeter of the circle.. ​

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

∘

2

1

∠CDA+

2

1

∠CBA=

2

1

×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.