Question

Prove that the perpendicular at the point of contact to the tangent to a circle passes through the center of the circle

Answer 1

Hey mate ...
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Soln--->

Let ,

O is the centre of the given circle.

A tangent PR has been drawn touching the circle at point P.

Draw QP ⊥ RP at point P, such that point Q lies on the circle.

∠OPR = 90°  (radius ⊥ tangent)

Also, ∠QPR = 90°  (Given)

∴ ∠OPR = ∠QPR

Now, above case is possible only when centre O lies on the line QP.

Hence, perpendicular at the point of contact to the tangent to a circle passes through the centre of the circle.

Hope it helps!!

Answer 2

Answer:

O is the centre of the given circle.

A tangent PR has been drawn touching the circle at point P.

Draw QP ⊥ RP at point P, such that point Q lies on the circle.

∠OPR = 90°  (radius ⊥ tangent)

Also, ∠QPR = 90°  (Given)

∴ ∠OPR = ∠QPR

Now, above case is possible only when centre O lies on the line QP.

Hence, perpendicular at the point of contact to the tangent to a circle passes through the centre of the circle.

Hope it helps!!