# Circles

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 Q1, Q2, Q3, ....Qn;

OQ1 > OP

OQ2 > OP

OQ3 > OP

OQn > 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 Î” POQ and Î” POR

OQ = OR (radii)

PO = PO (common side)

PQO = PRO (Right angle)

Hence; Î” POQ â‰ˆ Î” POR proved

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