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. Triangles are of three types based on their sides, they are:

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

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

What is Isosceles Triangle?

An Isosceles triangle is a triangle that has two equal sides. Also, the two angles opposite to the two equal sides are equal. In other words, we can say that “An isosceles triangle is a triangle which has two congruent sides”. 

Suppose in a triangle △ABC, if sides AB and AC are equal, then △ ABC is an isosceles triangle where ∠ B = ∠ C. The theorem that describes the isosceles triangle is “if the two sides of a triangle are congruent, then the angle opposite to them are also congruent”.

Angles of Isosceles Triangle

The two of the three angles of the isosceles triangle are equal in measure, which is opposite to the equal sides. Hence, one of the angles is unequal. Suppose, if the measure of an unequal angle is given to us, then we can easily find the other two angles by angle sum property.

Isosceles Triangle Properties

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 is 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, in an isosceles triangle, if two sides are congruent 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 in detail these three different types of an isosceles triangle.

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. Depends 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 to 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 act 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;

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, it is not possible to draw a triangle with more than two obtuse angles. 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:

where b is the base and h is the height of the triangle.

Perimeter of Isosceles Triangle

As we know the perimeter of any shape is the boundary of the shape. 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:

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

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. 

Practice Questions

1. What is the measure of the third angle, if the measure if equal angles of an isosceles triangle are 45°?
2. What is the perimeter of an isosceles triangle, if the measure of equal sides if 8.5 cm and the unequal side is 7 cm?
3. What is the area of a triangle if base = 10 cm and height = 6 cm?

Related Articles to Isosceles Theorem

• Isosceles Triangle, Equilateral Triangle, Scalene Triangle
• Isosceles Triangle Theorem
• Area of Isosceles Triangle
• Isosceles Triangle Formula

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