How can you prove a triangle is an equilateral triangle?

An Equilateral triangle is a triangle that has three equal sides and all three angles are also equal. It is also known as an equiangular triangle as the measure of every angle is equal to 60 degrees.

Answer: If three sides of a triangle are equal and the measure of all three angles is equal to 60 degrees then the triangle is an equilateral triangle. The distance formula can be used to prove that a triangle is an equilateral triangle.

Let' understand the explanation

Explanation:

We will use the figure given below to prove the triangle as an equilateral triangle.

Proof:

In the above figure,

Given, vertices of a triangle are A(X₁, Y₁), B(X₂, Y₂), and C(X₃, Y₃).

According to the properties of the equilateral triangle, the sides of an equilateral triangle should be equal in measure.

So, we use the distance formula:

d = √((x₂ - x₁)² + ( y₂ - y₁ )² to verify if all three sides are congruent and equal to each other

If AB = BC = CA.

⇒ √((x₂ - x₁)² + ( y₂ - y₁ )²) = √((x₃ - x₂)² + ( y₃ - y₂ )² = √((x₁ - x₃)² + ( y₁ - y₃ )²

If the given triangle satisfies the above condition then the triangle is an equilateral triangle.

Hence, if the given triangle has all three sides as congruent then the triangle is an equilateral triangle.