Question

show, that P(a,a),Q(-a,-a),R(-a√3,-a√3) form 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 } = √{4a^2 + 4a^2 } = √{8a^2 } = 2a√2

BC = √{(-a + a√3)^2 +(-a - a√3)^2 }

= √{a^2 + 3a^2 - 2a^2 *√3 + a^2 + 3a^2 + 2a^2 *√3 }

= √{8a^2 }

= 2a√2

CA = √{(a + a√3)^2 +(-a - a√3)^2 }

= √{a^2 + 3a^2 + 2a^2 *√3 + a^2 + 3a^2 - 2a^2 *√3 }

= √{8a^2 }

= 2a√2

Since, AB = BC = CA = 2a√2

So, the triangle ABC is an equilateral triangle.