Question:
show that the points (a, a), (-a,-a) and (- root3a , root3a) are the vertices of an equilateral triangle
Answer:
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.
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