# Two tangents from a point have equal length

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.

- 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°. - Both hypotenuses are the same. The two triangles share a hypotenuse, the line
**OP**, so they have to be equal in length. - 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.

## See also

- Parts of a circle
- Tangent and radius of a circle meet at 90°
- Two radii form an isosceles triangle
- Perpendicular bisector of a chord
- Angle at the centre of a circle is twice the angle at the circumference
- Angle in a semicircle is 90 degrees
- Angles in the same segment of a circle are equal
- Opposite angles in a cyclic quadrilateral add up to 180°

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