Tangents to a Circle | Moosa Academy

A tangent is a line that touches a circle at exactly one point, called the point of tangency. That single meeting point gives the tangent a clean right-angle relationship with the radius.

Theorem A tangent is perpendicular to the radius

At the point of tangency, the tangent line is perpendicular to the radius drawn to that point.

The radius OT and the tangent at T meet at 90°.

From a point inside the circle there is no tangent; on the circle, exactly one; outside, exactly two.

Theorem The length of a tangent

From an external point at distance d from the centre, the tangent length comes from the right triangle OTP by Pythagoras.

Since :

Note Tangent equations

For the circle , the tangent at a point on it is:

And a line is tangent to a circle exactly when its distance from the centre equals the radius r.

Reference Common tangents of two circles

Position of the circles	Common tangents
Separate (apart)	4
Touching externally	3
Intersecting	2
Touching internally	1
One inside the other	0

Example Three worked cases

Equation of a tangent

Circle

at the point

Use

⟹ the tangent is 3x + 4y = 25.

Length of a tangent

A point is

from the centre of a circle of radius

⟹ the tangent length is 12.

A round tower

A tower has radius 50 m; a person stands 130 m from its centre.

Shortest reach = tangent length

⟹ 120 m.

Summary

A tangent touches a circle at one point and is perpendicular to the radius there.
From an external point the tangent length is ; such a point has exactly two tangents.
The tangent to at is .
Two circles share 4, 3, 2, 1 or 0 common tangents, depending on how they sit.