Martin McBride, 2020-08-16

Tags circle tangent radius

Categories maths gcse geometry

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

If we draw a radius that meets the circumference at the same point, **the angle between the radius and the tangent will always be exactly 90°**.

This theorem is covered in this video on circle theorems:

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

We want to prove that the angle between the radius **AB** and the tangent **CD** is a right angle.

The way we will do this is to take some other point **P** on the tangent and prove that the line **AP** cannot be
perpendicular **CD**. If we prove that this cannot be true for any point **P**, then it follows that **AB** must
be perpendicular to **CD**.

We will start by *assuming* that **∠APB** is a right angle. We will then show that this leads to an impossibility and
so cannot be true.

Consider the triangle formed by points **A**, **B** and **P**. Suppose the angle of the triangle at **P** (that is**∠APB**) was a
right angle. This would mean that the line **AB** would be the *hypotenuse* of the triangle.

We know that the hypotenuse of a triangle is always longer than the other two sides, which means that **AB** must be longer than
**AP**. **But that cannot be true**. The line **AE** is the same length as the line **AB**, because each one is a radius of the circle.
**AP** is clearly longer than **AE**, therefore **AB** cannot be longer than **AP**. So, **the angle ∠APB cannot be a right angle**.

Since the line **AP** cannot be perpendicular to the tangent *for any P*, it follows that **AB** must be perpendicular to the tangent.

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