Question

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

Answer 1

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

To prove: ∠AOB = 90º .

Construction: Join OC.

Proof:
In ΔOPA and ΔOCA

OP = OC (Radii )

AP = AC (Tangents from point A)

AO = AO (Common )

ΔOPA ≅ ΔOCA (By SSS criterion)

$\therefore$ ∠POA = ∠COA .... (1) (By C.P.C.T)

Similarly, ΔOQB ≅ ΔOCB

∠QOB = ∠COB .......(2)

POQ is a diameter of the circle.

Hence, it is a straight line.

$\therefore$ ∠POA + ∠COA + ∠COB + ∠QOB = 180°

From equations (1) and (2), it can be observed that

2∠COA + 2∠COB = 180°

∴ ∠COA + ∠COB = 90°

∴ ∠AOB = 90°.