CBSE Class 10 Answered

In fig 1

Given:

QR is a tgt atQ to circle with centreP.

AQ is parellel toPR and AQ is a chord.AB is a diameter

To prove: BR is a tangent at B

Proof:   AQ  PR

 BPR=  PAQ (corressponding angle)  (1)

RPQ=PQA (alt. int.angles)  (2)

 PAQ= PQA (angles opp to equal sidesAP=PQ radii of same circle)  (3)

 BPR=QPR    (by 1,2 and 3)

In triangles QPR andBPR

PB=PQ  (radii of same circle)

PR=PR (comman)

QPR=BPR  (proved above)

TrianglePQR is to trianglePBR (by SAS cong condition)

 PQR= PBR  =90degrees.

Thus BR is a tangent at B