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Prove that the tangent at any …

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Prove that the tangent at any point of the circle is perpendicular to radius through the point of contact

Posted by Naina Goyal 6 years, 2 months ago

CBSE > Class 10 > Mathematics

1 answers

Yogita Ingle 6 years, 2 months ago

Given: A circle C (O, r) and a tangent AB at a point P.

To Prove: OP is perpendicular to AB.

Construction: Take any point Q, other than P, on the tangent AB. Join OQ.

Since, Q is a point on the tangent AB, other than the point of contact P, so Q will be outside the circle.

Let OQ intersect the circle at R.

Then, OQ=OR+RQ

⇒ OQ > OR

⇒ OQ > OP (OR = OP = radius)

Thus, OP
But, among all the line segments, joining the point O to a point on AB, the shortest one is the perpendicular from O on AB.
Hence, OP is perpendicular to AB.

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