# 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

## Join the GraphicMaths Newletter

Sign up using this form to receive an email when new content is added:

## Popular tags

adder adjacency matrix alu and gate angle area argand diagram binary maths cartesian equation chain rule chord circle cofactor combinations complex modulus complex polygon complex power complex root cosh cosine cosine rule cpu cube decagon demorgans law derivative determinant diagonal directrix dodecagon eigenvalue eigenvector ellipse equilateral triangle euler eulers formula exponent exponential exterior angle first principles flip-flop focus gabriels horn gradient graph hendecagon heptagon hexagon horizontal hyperbola hyperbolic function hyperbolic functions infinity integration by parts integration by substitution interior angle inverse hyperbolic function inverse matrix irrational irregular polygon isosceles trapezium isosceles triangle kite koch curve l system line integral locus maclaurin series major axis matrix matrix algebra mean minor axis nand gate newton raphson method nonagon nor gate normal normal distribution not gate octagon or gate parabola parallelogram parametric equation pentagon perimeter permutations polar coordinates polynomial power probability probability distribution product rule proof pythagoras proof quadrilateral radians radius rectangle regular polygon rhombus root sech set set-reset flip-flop sine sine rule sinh sloping lines solving equations solving triangles square standard curves standard deviation star polygon statistics straight line graphs surface of revolution symmetry tangent tanh transformation transformations trapezium triangle turtle graphics variance vertical volume of revolution xnor gate xor gate