# Theorems on Tangent to Circle

Tangent to a circle: A line which intersects a circle at any one point is called the tangent.

- There is only one tangent at a point of the circle.
- The tangent to a circle is perpendicular to the radius through the point of contact.
- The lengths of the two tangents from an external point to a circle are equal.

## Theorem 1

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

**Construction:** Draw a circle with centre O. Draw a tangent XY which touches point P at the circle.

**To Prove:** OP is perpendicular to XY.

Draw a point Q on XY; other than O and join OQ. Here OQ is longer than the radius OP.

OQ > OP

For every point on the line XY other than O, like Q_{1}, Q_{2}, Q_{3}, ……….Q_{n};

`OQ_1 > OP`

`OQ_2> OP`

`OQ_3 > OP`

`OQ_4 > OP`

Since OP is the shortest line

Hence, OP ⊥ XY proved

## Theorem 2

The lengths of tangents drawn from an external point to a circle are equal.

**Construction:** Draw a circle with centre O. From a point P outside the circle, draw two tangents P and R.

**To Prove:** PQ = PR

**Proof:** In Δ POQ and Δ POR

`OQ = OR` (radii)

`PO = PO` (common side)

`∠PQO = ∠PRO` (Right angle)

Hence; `Δ POQ ≅ Δ POR` proved