Question

Prove that the points (2 a, 4 a), (2 a, 6 a) and (2a+√3a,5a)are the vertices of an equilateral triangle.

Answer 1

For the points (2 a, 4 a), (2 a, 6 a) and (2a+√3a,5a) to be the vertices of an equilateral triangle , the distance between any two vertices should be same .

Distance between the points (2 a, 4 a) and (2a+√3a,5a)

D1 = √((2a-2a-√3a)² + (4a-5a)²)

= √(3a²+a²) = 2a

Distance between the points (2 a, 4 a) and (2 a, 6 a)

D2 = √((2a-2a)² + (4a-6a)²) = 2a

Distance between the points (2 a, 6 a) and (2a+√3a,5a)

D3 = √((2a-2a-√3a)² + (6a-5a)²)

= √(3a²+a²) = 2a

The value of D1 , D2 and D3 is same , so the the given points are the vertices of an equilateral triangle.