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 $\angle OTP = 90^\circ$ :

PT = \sqrt{d^{2} - r^{2}}

Note Tangent equations

For the circle $x^2 + y^2 = r^2$ , the tangent at a point $(x_1, y_1)$ on it is:

x x_1 + y y_1 = r^{2}

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

x^2 + y^2 = 25

at the point

(3, 4)

Use

x x_1 + y y_1 = r^2

3x + 4y = 25

⟹ the tangent is 3x + 4y = 25.

Length of a tangent

A point is

d = 13

from the centre of a circle of radius

r = 5

PT = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144}

⟹ 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

= \sqrt{130^2 - 50^2} = \sqrt{14400}

⟹ 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 $PT = \sqrt{d^2 - r^2}$ ; such a point has exactly two tangents.
The tangent to $x^2 + y^2 = r^2$ at $(x_1, y_1)$ is $x x_1 + y y_1 = r^2$ .
Two circles share 4, 3, 2, 1 or 0 common tangents, depending on how they sit.