What is an Isosceles Triangle?
A triangle is a two-dimensional shape with three sides, three angles, and three vertices. An isosceles triangle is a type of triangle in which two sides are of an equal length. In an isosceles triangle, the angles opposite to the equal sides are equal in measure.
In the given triangle ABC, the sides AB and AC are equal in length, that is, AB = AC. So, ΔABC is an isosceles triangle, and ∠B = ∠C.
Properties of an Isosceles Triangle
The properties of any geometric shape are based on its size, side measures, or angle measures. The properties of an isosceles triangle are as follows:
- Two sides are equal in length.
- The two equal sides of an isosceles triangle are known as ‘legs’ and the third side is known as the ‘base’.
- The angles opposite to the equal sides are equal in measure, that is, the two base angles are equal.
- The third angle, which is not equal to the two base angles, is known as the apex angle.
- The altitude of an isosceles triangle extends from the apex to its base.
- The altitude of an isosceles triangle bisects both the base and the apex angle into two equal parts.
- The altitude of an isosceles triangle divides the triangle into two congruent (equal) right triangles.
The Isosceles Triangle Theorem
The isosceles triangle theorem states that if two sides of an isosceles triangle are congruent, the angles opposite to them are also congruent.
So, in ΔABC shown in the image,
Given that AB = AC
Then, ∠ABC = ∠ACB
The converse of this theorem states that if two angles of an isosceles triangle are congruent, the sides opposite to them are also congruent.
So, in ΔABC shown in the image,
Given that ∠ABC = ∠ACB
Then, AB = AC.
Types of Isosceles Triangles
Isosceles triangles are classified into three different types:
- Isosceles acute triangle
- Isosceles right triangle
- Isosceles obtuse triangle
Isosceles acute triangle
An isosceles triangle in which all the three angles are less than 90°. For example, a triangle with angles measuring 55°, 55°, and 70° is an isosceles acute triangle.
Isosceles right triangle
An isosceles triangle in which one angle measures 90°. In an isosceles right triangle, the three angles are 45°, 45°, and 90°.
Isosceles obtuse triangle
An isosceles triangle in which one of the three angles is obtuse (lies between 90° and 180°). For example, a triangle with angles measuring 20°, 20°, and 140° is an example of an isosceles obtuse triangle.
The Area and Perimeter of an Isosceles Triangle
The area of an isosceles triangle is given as:
Area, A = \(\frac{1}{2}\) x base x height
= \(\frac{1}{2}\) x BC x AD
The perimeter of an isosceles triangle is given as:
Perimeter, P = AB + AC + BC
= 2AB + BC or, 2AC + BC or, (2 x leg) + Base [As AB = AC]
[Note: The altitude of a triangle is also known as its height.]
Solved Isosceles Triangle Properties Examples
Example 1: In an isosceles triangle, the length of the base is 6 cm and the length of the altitude drawn from the apex to the base is 8 cm. Find its area.
Solution:
The area of an isosceles triangle = \(\frac{1}{2}\) x base x height
= \(\frac{1}{2}\) × 6 × 8 [Substitute values]
= 24 [Simplify]
Hence, the area of the triangle is 24 \(cm^2\).
Example 2: Find the perimeter of an isosceles triangle if the base is 8 cm and the equal sides are 16 cm each.
Solution:
The two equal sides are the legs of the triangle.
The perimeter of an isosceles triangle is given as:
Perimeter = (2 x leg) + Base
= (2 x 16) + 8 [Substitute values]
= 40 [Simplify]
So, the perimeter of the isosceles triangle is 40 cm.
Example 3: What is the height of an isosceles triangle if its area is 30 \(cm^2\) and the base measures 5 cm?
Solution:
The area of an isosceles triangle = \(\frac{1}{2}\) x base x height
30 = \(\frac{1}{2}\) x 5 x height [Substitute values]
height = 12 [Simplify]
So, the height of the isosceles triangle is 12 cm.
Yes, this is because the altitude of an isosceles triangle divides the base into two equal parts.
The altitude of an isosceles triangle is the perpendicular line segment drawn from the vertex (or apex) of the triangle to the side opposite to it.
In an isosceles triangle, the angles opposite to the equal sides are equal, that is, the two base angles are congruent. This is known as the isosceles triangle base angle theorem.