# Isosceles Triangle

A triangle which has two of its sides equal in length.
Try this Drag the orange dots on each vertex to reshape the triangle. Notice it always remains an isosceles triangle, the sides AB and AC always remain equal in length
The word isosceles is pronounced "eye-sos-ell-ease" with the emphasis on the 'sos'. It is any triangle that has two sides the same length.

If all three sides are the same length it is called an equilateral triangle. Obviously all equilateral triangles also have all the properties of an isosceles triangle.

## Properties

• The unequal side of an isosceles triangle is usually referred to as the 'base' of the triangle.
• The base angles of an isosceles triangle are always equal. In the figure above, the angles ABC and ACB are always the same
• When the 3rd angle is a right angle, it is called a "right isosceles triangle".
• The altitude is a perpendicular distance from the base to the topmost vertex.

## Constructing an Isosceles Triangle

It is possible to construct an isosceles triangle of given dimensions using just a compass and straightedge. See these three constructions:

## Solving an isosceles triangle

The base, leg or altitude of an isosceles triangle can be found if you know the other two. A perpendicular bisector of the base forms an altitude of the triangle as shown on the right. This forms two congruent right triangles that can be solved using Pythagoras' Theorem as shown below.

### Finding the base

To find the base given the leg and altitude, use the formula: where:
L  is the length of a leg
A  is the altitude

### Finding the leg

To find the leg length given the base and altitude, use the formula: where:
B  is the length of the base
A  is the altitude

### Altitude

To find the altitude given the base and leg, use the formula: where:
L  is the length of a leg
B  is the base

## Interior angles

If you are given one interior angle of an isosceles triangle you can find the other two.

For example, We are given the angle at the apex as shown on the right of 40°. We know that the interior angles of all triangles add to 180°. So the two base angles must add up to 180-40, or 140°. Since the two base angles are congruent (same measure), they are each 70°.

If we are given a base angle of say 45°, we know the base angles are congruent (same measure) and the interior angles of any triangle always add to 180°. So the apex angle must be 180-45-45 or 90°.