prove that the intercept of a tangent between 2 parallel tangents to a circle subtends a right angle at the center of the circle.

Asked by anukeshd | 14th Mar, 2013, 07:58: PM

Expert Answer:

Let XY and X 'Y ' be the two parallel tangents and AB intersect them at A and B reapectively such that AB subtends angle AOB at the centre.

Join OC

In ?OPA and ?OCA,

OP = OC (Radii of the same circle)

AP = AC (Tangents from point A)

AO = AO (Common side)

?OPA ?OCA (SSS congruence criterion)

Therefore, ?POA = ?COA … (i)

Similarly, ?OQB ?OCB 

?QOB = ?COB … (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

2?COA + 2 ?COB = 180

?COA + ?COB = 90

?AOB = 90

Answered by  | 14th Mar, 2013, 09:33: PM

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