## Summary

## Table of Contents[Show]

A straight line intersects a circle at one or two point. The tangent to a circle is a line that touches the circle at one point. Secant intersects the circle at two points.

**Point of contact:**

The point at which the straight line touches the circle is called the point of contact or point of tangency.

**Some properties of tangents to a circle:**

- Infinite number of tangents can be drawn to a circle but only one tangent can be drawn at any given point on a circle.
- From an external point to can draw two tangents of equal length.
- The radius of the circle is perpendicular to the tangent at its point of contact and the tangents drawn at the extremities of the diameter of a circle are parallel.

**Theorem:**

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

Given: A tangent AB with point of contact P.

To prove: OP ⊥ AB

Proof:

Consider point C on AB other than P.

C must lie outside the circle. (∵ A tangent can have only one point of contact with the circle)

OC > OP (∵ C lies outside the circle)

This is true for all positions of C on AB.

Thus, OP is the shortest distance between point P and line segment AB.

Hence, OP ⊥ AB.

**Theorem:**

Tangents drawn to a circle from an external point are equal in length.

Given: Two tangents AB and AC from an external point A to points B and C on a circle.

To prove: AB = AC

Construction: Join OA, OB and OC.

Proof:

In triangles OAB and OAC,

∠OBA = 90

^{o}(Radius OB ⊥ Tangent AB at B)

∠OCA = 90

^{o }(Radius OC ⊥ Tangent AC at C)

In triangles OBA and OCA,

∠OBA = ∠OCA = 90

^{o }

OB = OC (Radii of the same circle)

OA = OA (Common side)

Thus, ΔOBA ≅ ΔOCA (RHS congruence rule)

Hence, AB = AC (Corresponding sides of congruent triangles)

The tangents drawn at the extremities of the diameter of a circle are parallel.