# Circle

## Theorems on Circle

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

1. There is only one tangent at a point of the circle.
2. The tangent to a circle is perpendicular to the radius through the point of contact.
3. 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 Q1, Q2, Q3, ……….Qn;

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