Question

In the figure, XP and XQ are tangents from X to the circle with centre O. R is a point on the circle. Prove that XA + AR = XB + BR.

Answer 1

Given:

XP and XQ are tangents from an external point X to a circle with centre O.

RRR is a point on the circle.

The line XR meets the tangents at points A and B respectively.

We have to prove:

XA + AR = XB + BR

Tangents drawn from an external point to a circle are equal in length.

XP = XQ     ...(1)

Segment addition on straight lines

XP = XA + AP

XQ = XB + BQ

Substitute from (1)

XA + AP = XB + BQ     ...(2)

Equal tangent segments from a point on the circle

Since R lies on the circle:

AP and AR are tangents from point A

BQ and BR are tangents from point B

AP = AR and BQ = BR

XA + AR = XB + BR

Hence Proved.