Prove that the perpendicular at the point of contact to the tangent to

1.	Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
Answer» Data: Perpendicular at the point of contact to the tangent to a circle passes through the centre. PQ is the tangent to circle with centre ’O’. Let the perpendicular drawn at the point P is ∠RPQ. ∠RPQ = 90° …………. (i) Radius drawn to circle at the point of contact is perpendicular. ∴ ∠OPQ = 90° …………. (ii) From eqn. (i) and eqn. (ii), we have ∠RPQ = ∠OPQ = 90° . This is contradiction, because ∠RPQ is the part of ∠OPQ. ∴ ∠RPQ < ∠OPQ ∴ ∠RPQ ≠∠OPQ. ∴ The perpendicular at the point of contact to the tangent to a circle passes through the centre.

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

Answer»

Data: Perpendicular at the point of contact to the tangent to a circle passes through the centre.

PQ is the tangent to circle with centre ’O’.

Let the perpendicular drawn at the point P is

∠RPQ. ∠RPQ = 90° …………. (i)

Radius drawn to circle at the point of contact is perpendicular.

∴ ∠OPQ = 90° …………. (ii)

From eqn. (i) and eqn. (ii), we have

∠RPQ = ∠OPQ = 90° .

This is contradiction, because

∠RPQ is the part of ∠OPQ.

∴ ∠RPQ < ∠OPQ

∴ ∠RPQ ≠∠OPQ.

∴ The perpendicular at the point of contact to the tangent to a circle passes through the centre.

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