Prove that the perpendicular at the point of contact . . .

Question :  Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre. [NCERT]

Doubt by Saumya G.

Solution : 

Given : A circle with centre O. PQ is a tangent at point of contact P. AP⊥PQ such that P is not passing through the centre. 

To Prove : Perpendicular AP is passing through the centre O of the circle. 

Construction : Join OP.

Proof : 

AP⊥PQ (Given)

∴ ∠APQ = 90°— (1) 

OP⊥PQ [Tangent at any point of the circle is perpendicular to the radius through the point of contact] 
∴ ∠OPQ = 90°— (2) 

From equation (1) and (2)

∠APQ = ∠OPQ

But this is only possible when AP lies on OP.

Hence, we can say that the AP must pass through the centre O. 

Hence Proved.