Two radii form an isosceles triangle

A triangle formed by two radii of a circle is always an isosceles triangle.

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Isosceles triangles

An isosceles triangle is a triangle that has two equal sides, like this:

The two angles at the base of an isosceles triangle are always equal.

Two radii of a circle

Two radii form the two equal sides of a triangle. The third side is formed by the chord that joins the ends of the two radii.

Since it is an isosceles triangle, it follows that the angles between the two radii and the chord that forms the third side of the triangle are also equal.

Proof

A circle is a set of points that are all the same distance from the centre.

A radius is a line drawn from the centre to any point on the circle.

Since every point on the circle is the same distance from the centre, every radius is the same length no matter where it is drawn.

Two sides of the triangle are radii of the circle and therefore have the same length. Any triangle with two equal sides is an isosceles triangle, by definition.

Since it is an isosceles triangle, the two angles A and B are also equal. This fact is sometimes useful, for example:

It might not be immediately obvious what the angle x is. But if we remember that the triangle is an isosceles triangle we can see very easily that x is equal to the other angle at the base, so it is 70 degrees.

These resources are designed to support GCSE Maths students. They cover a range of useful mathematical topics but are not intended to provide a complete GCSE course or follow any particular exam board's specification. More...

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