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prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact
Asked by muskanmahek2411 | 08 Oct, 2021, 10:48: PM
answered-by-expert Expert Answer
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.

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.Gi

Answered by Renu Varma | 23 Oct, 2021, 06:28: PM

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