Question

Show that the points (a) (_ a_ a ) and _ root 3 ,a,root3 a ) are the vertices of an equilateral triangle

Answer 1

Given points are A(a, a), B(-a,-a) and C(-a√3 , a√3)

Now, AB = √{(a + a)2 +(a + a)2 } = √{(2a)2 + (2a)2 } = √{4a2 + 4a2 } = √{8a2 } = 2a√2

BC = √{(-a + a√3)2 +(-a - a√3)2 }

= √{a2 + 3a2 - 2a2 *√3 + a2 + 3a2 + 2a2 *√3 }

= √{8a2 }

= 2a√2

CA = √{(a + a√3)2 +(-a - a√3)2 }

= √{a2 + 3a2 + 2a2 *√3 + a2 + 3a2 - 2a2 *√3 }

= √{8a2 }

= 2a√2

Since, AB = BC = CA = 2a√2

So, the triangle ABC is an equilateral triangle.