Question

Show that the points (a a) (-a -a)( -√3a,√3a) are the verticies of an equilateral triangle.

Answer 1

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Answer 2

Proved that the triangle is an equilateral triangle.

Step-by-step explanation:

We have to show that the triangle with vertices A(a,a), B (-a,-a) and C(-√3a,√3a) is an equilateral triangle.

Now, the length of AB = $\sqrt{(a - (- a))^{2} + (a - (- a))^{2}} = \sqrt{8a^{2}} = 2\sqrt{2}a$ units.

Again, the length of BC =  $\sqrt{(-a + \sqrt{3}a )^{2} + (-a - \sqrt{3}a )^{2}} = \sqrt{2(a^{2} + (\sqrt{3}a )^{2})} = 2\sqrt{2}a$ units.

{Since we know that (a + b)² + (a - b)² = 2(a² + b²)}

And, the length of CA = $\sqrt{(a + \sqrt{3}a )^{2} + (a - \sqrt{3}a )^{2}} = \sqrt{2(a^{2} + (\sqrt{3}a )^{2})} = 2\sqrt{2}a$ units.

Therefore, all the sides of the triangle are equal in length and hence the triangle is an equilateral triangle. (Answer)