# Tangent and radius of a circle meet at 90°

Categories: 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:

## Intuitive argument

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

When we draw a radius vertically (circle on the left), the diagram is symmetrical. The left half of the circle is a mirror image of the right half.

If we add a tangent (circle on the right), then the radius meets the tangent forming an angle **a** on the left and **b** on the right.

Due to symmetry, there is no reason to assume that either **a > b** or **a > b**. Intuitively it seems reasonable to assume that the two angles are equal, and since they add to make a straight line they would each be 90 degrees.

This is not a proof, it is just a way of convincing ourselves that it is probably true. The proof is next.

## Proof

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 **OP** and the tangent **AB** is a right angle.

We will use *proof by contradiction*. This means that we will start by assuming that the statement is *not true*. This will lead to a contradiction, proving that it is impossible for the statement to be not true, which means it must be true.

So we start with the assumption that the line **OP** does not meet the line **AB** at a right angle.

There must be some other line from **O** that meets the line **AB** at a right angle. It meets the line **AB** at some point **C**. Here is a diagram:

Since the circle only touches the tangent in one place, point **P**, it follows that the line **OC** must cross the circle at some point. We will call that point **D**.

We have deliberately chosen point **C** such that the angle **∠OCA** is a right angle. So we can now draw just the triangle on its own:

One thing to notice here is that the lines **OD** and **OP** are both radii of the circle, so they both have equal lengths. Since the line **OC** clearly longer than **OD**, it must also be longer than **OP**

But this is an impossible triangle, because:

- Angle
**C**is the right angle, so side**OP**must be the hypotenuse. - The hypotenuse,
**OP**of a triangle is always the longest side. - But we know that
**OC**is always longer than**OP**.

This is a contradiction. It means that it is impossible for line **OC** to meet line **AB** at a right angle.

Now **C** can be placed anywhere on the line **AB** *except* at position **P**. But where ever we place it, the angle **∠ACB** cannot be a right angle.

There must be a line from **O** that meets **AB** at a right angle, and it can't be at any position **C** that isn't **P**, so it must be at position **P**.

**OP** is the radius, and **AB** is the tangent, and we have proved that they meet at a right angle.

## See also

- Parts of a circle
- 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°
- Two tangents from a point have equal length

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