Two tangents from a point have equal length

By Martin McBride, 2022-12-22
Categories: gcse geometry

A tangent is a line that just touches the circle at a single point on its circumference.

If we take any point P outside the circle, we can draw two tangents. Those two tangents will have equal lengths.

This means that the length PA and the length PB will always be equal.

Here is a video on the topic:

Intuitive argument

We can use symmetry to provide an intuitive argument as to why this might be true.

The diagram on left shows a circle with a point P directly below it. This diagram is symmetrical - the left and right halves are identical mirror images of each other.

We then draw two tangents (diagram on the right).

Due to symmetry, there is no reason to assume that one tangent should be longer than the other. Intuitively it seems reasonable to assume that the two tangents are equal.

This is not a proof, but it indicates that the tangents are probably equal. The proof is given next.

Proof

You aren't required to learn this proof for GCSE, it is just here for information.

We want to prove that PA and PB are equal.

We will prove this by proving that triangles OPA and OPB are congruent:

We will use three facts to prove that the triangles are congruent.

1. They are both right-angled triangles. For the triangle OPA, the side OA is a radius and the side PA is a tangent. The tangent and radius circle theorem tells us that a radius and tangent meet at 90°, so the angle at A is 90°. The same is true for the triangle OPB and the angle at B is also 90°.
2. Both hypotenuses are the same. The two triangles share a hypotenuse, the line OP, so they have to be equal in length.
3. The two triangles also have a side of the same length. The side OA of triangle OPA is a radius. The side OB of triangle OPB is also a radius. So both triangles have an equal side.

According to the RHS rule of congruent triangles, two right-angled triangles where the hypotenuse is equal and any side is equal are congruent.

This means that the third side of each of the two triangles are equal, PA equals PB, which is what we set out to prove.