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

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)²  =  (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