Prove that the intercept of a tangent between two parallel

Question:

Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at the centre.

Solution:

Given: XY and XY at are two parallel tangents to the circle with centre O and AB is the tangent at the point C, which intersects XY at A and XY at B.

To Prove: AOB = 90°

Construction: Let us joint point O to C.

Proof:

In ΔOPA and ΔOCA, we have

OP = OC (Radii of the same circle)

AP = AC (Tangents from point A)

AO = AO (Common side)

ΔOPA  ΔOCA (SSS congruence criterion)

Therefore, POA = COA ……(i) (C.P.C.T)

Similarly, ΔOQB  ΔOCB ……(ii)

Since POQ is a diameter of the circle, it is a straight line.

Therefore, POA + COA + COB + QOB = 180°

From equations (i) and (ii), it can be observed that

2COA + 2COB = 180°

∴ ∠COA + COB = 90°

So, AOB = 90°.

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