Question

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

Answer 1

Let O be the centre of the given circle.
AB is the tangent drawn touching the circle at A.
Draw AC ⊥ AB at point A, such that point C lies on the given circle.
∠OAB = 90° (Radius of the circle is perpendicular to the tangent)
Given ∠CAB = 90°
∴ ∠OAB = ∠CAB
This is possible only when centre O lies on the line AC.
Hence, perpendicular at the point of contact to the tangent to a circle passes through the centre of the circle.