
Isosceles Triangle
From Greek: isos  "equal" , skelos  "leg"
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 "eyesosellease" 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.
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


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 18040, 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 1804545 or 90°.
Other triangle topics
General
Perimeter / Area
Triangle types
Triangle centers
Congruence and Similarity
Solving triangles
Triangle quizzes and exercises
(C) 2009 Copyright Math Open Reference. All rights reserved

