Isosceles Triangle

An isosceles triangle is a type of triangle that has any two sides equal in length. The two angles of an isosceles triangle, opposite to equal sides, are equal in measure. In geometry, triangle is a three-sided polygon that is classified into three categories based on its sides, such as:

• Scalene triangle (All three sides are unequal)
• Isosceles triangle (Only two sides are equal)
• Equilateral triangle (All three sides are equal)

In this article, we will learn the properties and formulas related to the isosceles triangle, in detail, along with examples.

Properties of Isosceles Triangle

1. As the two sides are equal in this triangle, the unequal side is called the base of the triangle
2. The angles opposite to the two equal sides of the triangle are always equal
3. The altitude of an isosceles triangle is measured from the base to the vertex (topmost) of the triangle
4. A right isosceles triangle has a third angle of 90 degrees

Isosceles Triangle Theorem

As per the theorem, if two sides are congruent in an isosceles triangle, then the angles opposite to the two sides are also congruent. 

Alternatively, if two angles are congruent in an isosceles triangle, then the sides opposite to them are also congruent. 

In the above triangle ABC, 

AB = AC

∠ABC = ∠ADC

Types of Isosceles Triangle

Generally, the isosceles triangle is classified into different types, namely, 

• Isosceles acute triangle
• Isosceles right triangle
• Isosceles obtuse triangle

Now, let us discuss these three different types of an isosceles triangle in detail.

Isosceles Acute Triangle

As we know that the different dimensions of a triangle are legs, base, and height. All the isosceles triangle has an axis of symmetry along the perpendicular bisector of its base. Depending on the angle between the two legs, the isosceles triangle is classified as acute, right and obtuse. The isosceles triangle can be acute if the two angles opposite the legs are equal and are less than 90 degrees (acute angle).

Isosceles Right Triangle

A right isosceles triangle has two equal sides, wherein one of the two equal sides acts as perpendicular and another one as a base of the triangle. The third side, which is unequal, is termed the hypotenuse. Therefore, we can apply here the famous Pythagoras theorem, where the square of the hypotenuse is equal to the sum of the square of base and perpendicular.

Suppose the sides of the right isosceles triangle are a, a, and h, where a  is the two equal sides, and h is the hypotenuse, then;

h = √(a2 + a2) = √2a2 = a√2or h = √2 a

Isosceles Obtuse Triangle

We know that the obtuse triangle is a triangle in which one of its angles is greater than 90 degrees (right angle). Also, drawing a triangle with more than two obtuse angles is impossible. We know that the obtuse triangle can be a scalene triangle or isosceles triangle. Therefore, the isosceles obtuse triangle is a triangle which has two equal sides with an obtuse angle.

Isosceles Triangle Formulas

We know that an isosceles triangle is a two-dimensional shape with three sides. The measures to compute the isosceles triangle are the area and perimeter. Now, let us discuss the area and the perimeter of the isosceles triangle in detail. 

Area of Isosceles Triangle

The area of an isosceles triangle is defined as the region occupied by it in the two-dimensional space. Generally, the isosceles triangle is half the product of the base and height of an isosceles triangle. The formula to calculate the area of an isosceles triangle is given by:

The area of an isosceles triangle A = ½ × b × h Square units

where b is the base and h is the triangle’s height.

Perimeter of Isosceles Triangle

As we know, the perimeter of any shape is the shape’s boundary. Similarly, the perimeter of an isosceles triangle is defined as the sum of the three sides of an isosceles triangle. The perimeter of an isosceles triangle can be found if we know its base and side. The formula to calculate the perimeter of the isosceles triangle is given by:

Perimeter of an isosceles Triangle, P = 2a + b units

where ‘a’ is the length of the two equal legs of an isosceles triangle and b is the base of the triangle.

Isosceles Triangle Altitude

When an altitude is drawn to the base of the isosceles triangle, it bisects the vertex angle. As it bisects the base, the two congruent triangles are created. The altitude of the triangle forms the required right angle and the altitude becomes the shared legs. Also, the congruent legs of a triangle become the congruent hypotenuse. Therefore, the altitude of drawn to the base of the isosceles triangle bisects the base.

Solved Examples on Isosceles Triangle

Example 1: Find the area of an isosceles triangle given its height as 6 cm and base as 4 cm?

Solution:

Given that,

Base = 4 cm and height = 6 cm

We know that the area of an isosceles triangle is ½ × b × h square units

Now, substitute the base and height value in the formula

Area of an isosceles triangle is ½ × b × h

A = ½ × 4 × 6 = 12 cm2

Therefore, the area of an isosceles triangle is 12 cm2.

Example 2: Find the perimeter of an isosceles triangle, with a side 6 cm and base 4 cm.

Solution:

Given: Base = 4 cm

Length of the two equal arms = 6 cm

We know that the formula to calculate the perimeter of an isosceles triangle is  P = 2a + b units

Now, substitute the values in the perimeter formula, we get

P = 2(6) + 4 = 12 + 4 = 16 cm

Hence, the perimeter of an isosceles triangle is 16 cm.