Isosceles Triangle
A triangle having two equal sides is known as an isosceles triangle. The two equal sides are called legs, and the third side is known as the base of the isosceles triangle. The angle between the legs is called the vertex angle, and the two angles adjacent to the base are called base angles.
Area and Perimeter of an Isosceles Triangle
The formula for the area of an isosceles triangle is written as:
Area (A) = \(\frac{1}{2}\) x b x h
Here, ‘b’ refers to the base and ‘h’ refers to the height of the triangle.
The perimeter of an isosceles triangle is determined by applying the formula:
Perimeter (P) = 2a + b
Here, ‘a’ is the length of the equal sides of an isosceles triangle, and ‘b’ is the length of the base.
Properties of an Isosceles Triangle
The properties of an isosceles triangle are:
- An isosceles triangle has two equal sides.
- An isosceles triangle has two equal angles.
- An isosceles triangle has two acute angles.
- The sum of three angles of an isosceles triangle is 180°.
Rapid Recall
Solved Examples
Example 1:
The base of an isosceles triangle is 20 centimeters, and the length of each of its legs is 28 centimeters. Calculate its perimeter.
Solution:
AB = AC = a = 28 cm and BC = b = 20 cm.
The formula for finding the perimeter is:
Perimeter (P) = 2a + b
= 2(28) + 20 [Substitute 28 for ‘a’ and 20 for ‘b’]
= 56 + 20 [Simplify]
= 76 cm
Hence, the perimeter of the isosceles triangle is 76 centimeters.
Example 2:
Winnie made a sandwich for her brother. She cut the sandwich into a triangular slice such that the base length of the triangle is 15 centimeters and the height is 20 centimeters. Determine the area of the sandwich.
Solution:
Winnie cut the sandwich into a triangular slice, having a base length (b) of 15 cm and a height (h) of 20 cm.
Area (A) = \(\frac{1}{2}\) × b × h [Formula for the area of the triangle]
= \(\frac{1}{2}\) × 15 × 20 [Substitute 15 for ‘b’ and 20 for ‘h’]
= \(\frac{300}{2}\) [Simplify]
= 150 cm\(^2\)
Therefore, the area of the sandwich is 150 square centimeters.
Example 3:
The area of a triangle is 52 square inches, and the base length is 8 inches. Calculate the height of the triangle.
Solution:
Area (A) = 52 in
base (b) = 8 in
Area (A) = \(\frac{1}{2}\) × b × h [Area of a triangle formula]
52 = \(\frac{1}{2}\) × 8 × h [Substitute 52 for A and 8 for b]
52 = 4 × h [Multiply]
\(\frac{52}{4}\) = h [Simplify]
13 = h
Thus, the height of the given triangle is 13 inches.
If two sides of a triangle are equal, the triangle is said to be an isosceles triangle. Consider a triangle with sides AB, BC, and CA. The triangle is isosceles if it fulfills one of these conditions: AB = BC, or BC = CA, or CA = AB.
Yes, by knowing the two equal angles, we can easily subtract the sum of angles from 180°, as the sum of all angles of a triangle is equal to 180°.
Some of the examples of an isosceles triangle are the roof of a house, a slice of pizza, clothes hangers, traffic signs, and so on.
There are three types of isosceles triangles:
- Isosceles acute angle triangles
- Isosceles right angle triangles
- Isosceles obtuse angle triangles