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

Abhinav Verma
From Vivekanand School D Block Anand Vihar, added an answer, on 23/2/11

2077 helpful votes in Math
Hi Rida

Draw a circle with centre O.

draw a tangent PR touching circle at P.

Draw QP perpendicular to RP at point P, Qp lies in the circle.

Now, angle OPR = 90 degree (radius perpendicular to tangent)

also angle QPR = 90 degree (given)

Therefore angle OPR = angle QPR. This is possible only when O lies on QP. Hence, it is proved that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

draw a tangent PR touching circle at P.

Draw QP perpendicular to RP at point P, Qp lies in the circle.

Now, angle OPR = 90 degree (radius perpendicular to tangent)

also angle QPR = 90 degree (given)

Therefore angle OPR = angle QPR. This is possible only when O lies on QP. Hence, it is proved that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

Cheers

Best of luck for your exams

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