Isosceles Triangle: Definition, Properties, and Examples

Triangles with at least two equal sides are referred to as isosceles triangles. The word isosceles originates from Greek words isos meaning equal, and skelos, meaning leg. Therefore, an isosceles triangle is defined by having two sides of the same length, which are often called legs, and a third side called the base, which is either longer or distinct from the legs.

The triangle has three types based on its sides in Geometry, such as:

• Scalene Triangle
• Isosceles Triangle
• Equilateral Triangle

We will confine ourselves to only the isosceles triangle in this blog. We will dive deep into the world of isosceles triangles, exploring their properties, types, and many examples.

Definition of Isosceles Triangle

An isosceles triangle is a particular type of triangle in which two sides have an equal length. These two sides are known as the legs of the triangle and the third side is referred to as the base. The angles opposing the legs have the equal measurement. Therefore, an isosceles triangle has two congruent and one different angle. The congruent angles are always opposite the sides that have the same length.

Useful Properties of an Isosceles Triangles

Isosceles triangles possess particular characteristics that distinguish them from other types of triangles. Here are some notable characteristics of isosceles triangles:

• Two congruent sides are the most defining characteristic of an isosceles triangle.
• An isosceles triangle always has congruent angles at the base. This property is a direct consequence of the equal side lengths.
• The angle opposite the base in an isosceles triangle is generally larger than the base angles.
• The line drawn from the vertex angle to the midpoint of the base bisects the triangle into two congruent right triangles.

Isosceles Triangles Types

Isosceles triangles can be categorized into different types based on their unique properties. Here are the main types of isosceles triangles:

• Acute Isosceles Triangle
• Obtuse Isosceles Triangle
• Right Isosceles Triangle

Acute Isosceles Triangle:

In this type, all three interior angles are acute angles, which means they are less than 90 degrees. The two base angles and the vertex angle are all acute.

Obtuse Isosceles Triangle:

An obtuse isosceles triangle has one obtuse angle, which is an angle greater than 90°, typically occurring at the vertex. The other two angles are acute and equal, forming the base angles.

Right Isosceles Triangle:

A right isosceles triangle is characterized by one right angle, which measures 90 degrees, typically located at the vertex. The other two angles are equal and acute, known as the base angles.

Formula to Determine the Area and Perimeter of Isosceles Triangles

• This formula can be used to find the area of an isosceles triangle:

Area = (1/2) × base (b) × height (h)

• You can use the following formula to determine the perimeter of an isosceles triangle:

Perimeter = 2a + base (b)

Here, a represents the length of the identical sides of the isosceles triangle, while b indicates the length of the third side, which is not equal to the other two.

Angle of Isosceles Triangles

Angles in an isosceles triangle are related in a specific way due to the symmetry of the triangle. You can easily determine the measures of the other two angles using the angle sum property when one of the unequal angles is given.

For Example:

Consider an isosceles triangle where one of the unequal angles measures 80° and the two equal angles are denoted by y. According to the angle sum property:

80° + y + y = 180°

Now, simplify the equation:

80° + 2y = 180°

Next, subtract 80° from both sides to isolate 2y:

2y = 180° - 80°

2y = 100°

Finally, divide both sides by 2 to find the value of y:

y = 100° / 2

y = 50°

Thus, the two equal angles in the isosceles triangle both measure 50 degrees.

Solved Examples of isosceles Triangle

Example 1:

Evaluate the area of an isosceles triangle with a height of 8 cm and a base of 5 cm.

Solution:

Given Data:

Height (h) = 8 cm

Base (b) = 5 cm

We can find the area (A) of the isosceles triangle using the formula:

A = 1/2 × b × h

Substitute the given values:

A = 1/2 × 5 cm × 8 cm

= 20 cm²

∴ The isosceles triangle has an area of 20 square centimeters.

Example 2:

Evaluate the perimeter of an isosceles triangle with two equal sides measuring 9 centimeters each and a base of 7 cm.

Solution:

Given Data:

Length of equal sides (a) = 9 cm

Base (b) = 7 cm

The equation to evaluate the perimeter (P) of an isosceles triangle is:

P = 2a + b

Substitute the given values:

P = 2 × 9 cm + 7 cm

= 18 cm + 7 cm

= 25 cm

Therefore, the perimeter (P) of this isosceles triangle measures 25 cm.

Conclusion

In this article, we have explored the world of isosceles triangles, delving into their definition, properties, types, and various theorems related to them. We have learned how to determine the perimeter and area of isosceles triangles using appropriate formulas.

Additionally, we have demonstrated how to find the angles in isosceles triangles using the angle sum property. Through solved examples, we have illustrated practical applications of these concepts.