Question

Prove that the tangent at any point of a circle is perpendicular to theradius through the point of contact.

Answer 1

Hey mate..
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Proof:--
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Let, PT be a tangent to the circle with centre O and the point A be the point of contact. B is another point taken on PT. O,A and O,B are joined. Since the line PT touches the circle only at the point A , so it cannot cut the circle at any other point. Therefore , of all the points lying on PT except the point A will lie outside the circle. So, whatever be the point B on PT ( OB > OA ) i.e. of all the line segments drawn from the point O to PT , OA is the shortest.

which means , OA is perpendicular to PT

Thus, the tangent at any point of a circle is perpendicular to theradius through the point of contact.

Hope it helps !!