# Isosceles Triangle

## Definition

Isosceles triangle is a figure where two sides are of equal length and two angles are equal.

## Properties

1. Two equal (congruent) sides; in ∆ABC, AB and AC are two congruent sides
2. One line of symmetry
3. The two angles opposite to the equal sides are equal (isosceles triangle base angle theorem). In ∆ABC, since AB = AC, ∠ABC = ∠ACB
4. The Altitude, AE bisects the base and the apex angle into two equal parts, forming two congruent right-angled triangles, ∆AEB and ∆AEC

## Types

Isosceles triangles are classified into three types: 1) acute isosceles triangle, 2) obtuse isosceles triangle, and 3) right isosceles triangles.

The differences between the types are given below:

## Formulas

### Area

Area (A) = ½ (b × h), where b = base and h= height

Problem: Finding the area of an isosceles triangle when only  THREE SIDES are known

Find the area of an isosceles triangle whose three sides measure 5 cm, 5 cm, and 6 cm

Solution:

Here we will use the Pythagoras theorem to calculate the height of the triangle,
(hypotenuse)=  (height)2   + (base)2
Let, a = hypotenuse,
b = base,
h = height
The equation becomes,
h = √a2 – (b/2)2
In this triangle, a = 5 cm, b = 6 cm
Hence, h = √52 – (6/2)2 cm
= √ 25 – 9 cm
= 4 cm
Now, A = ½ (b x h)
= ½ (6 x 4) cm2
= 12 cm2

### Perimeter

We know, P = a+ b +c, where a, b, c are the measures of three sides

Since in an isosceles triangle two sides are equal i.e., a = c

Hence, P = 2a + b