Question

QR is a tangent at Q, PR ||AQ, where AQ is a chord through A and P is a centre, the point of the diameter AB. Prove that BR is tangent at B.

Answer 1

REF.Image

Given : P is center of circle OR QR is tangent at Q. AB is diameter and AQ||PR

To prove : BR is a tangent to circle.

Proof : In $\Delta APQ$

AP=PQ [ radius]

$\angle 1=\angle 2$ [ equal $\angle S$ have equal sides opposite and uice versa]

Given AQ||PR

$\angle 1=\angle 3$ [Pair of corresponding $\angle S$]

$\angle 2=\angle 4$ [Pair of alternate $\angle S$]

Thus $[\angle 3=\angle 4]$

$\therefore \angle 1=\angle 2$

In $\Delta PQR$ and $\Delta PBR$

PQ=PB [radius]

$\angle 3=\angle 4$ [proved]

PR=PR [common]

$\therefore \Delta PQR\cong \Delta PBR$ [SAS]

and $\angle PQR=\angle PBR={90}^{\circ }$ [CPCT]

Hence BR is tangent to circle.