Theorem
The two tangents from an external point
Geometry • Circle • Tangents
Statement
A tangent is perpendicular to the radius at its point of contact, and the two tangents drawn to a circle from an external point are equal in length; the line to the centre bisects both the angle between the tangents and the angle between the two radii, so each tangent has length .
How to use it
- 1Halve the angle between the tangents before any trigonometry: each half sits in a right triangle with legs and the tangent length.
- 2If the tangents meet at , the tangent length is and the distance to the centre is .
- 3Equal tangents from each vertex is what makes the four sides of a circumscribed polygon decomposable into pairs of equal pieces.
Worked example
A point lies cm from the centre of a circle of radius cm. How long is a tangent from it?
Step-by-step
- 1The radius to the point of contact is perpendicular to the tangent.
Why: So the radius, the tangent and form a right triangle with as hypotenuse.
- 2cm
Answer
cm
Watch out
The angle in the right triangle is half the angle between the tangents, not the whole of it — using the full angle is the standard error. The tangent length is not itself, and it is defined only for a point outside the circle.