Question

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

Answer 1

9th one. Hope this helps you....

Answer 2

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=90

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Construction:Join OC

Proof:In △OPA and △OCA

OP=OC (radii)

AP=AC (Tangents from point A)

AO=AO (Common )

△OPA≅△OCA (By SSS criterion)

∴,∠POA=∠COA .... (1) (By C.P.C.T)

OQ=OC (radii)

BQ=BC (Tangents from point A)

BO=BO (Common )

△OQB≅△OCB

∠QOB=∠COB .......(2)

POQ is a diameter of the circle.

Hence, it is a straight line.

∴,∠POA+∠COA+∠COB+∠QOB=180

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

2∠COA+2∠COB=180

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∴∠COA+∠COB=90

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∴∠AOB=90

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