The points (a, a), (−a, a) and $(- \sqrt{3} a, \sqrt{3} a)$ form the vertices of

(a) an equilateral triangle
(b) a scalene triangle
(c) an isosceles triangle
(d) a right triangle

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Solution

(a) an equilateral triangle
Let A(a, a), B(−a, −a) and $C (- \sqrt{3} a, \sqrt{3} a)$ be the given points. Then,
$A B = \sqrt{{(- a - a)}^{2} + {(- a - a)}^{2}} = \sqrt{{(- 2 a)}^{2} + {(- 2 a)}^{2}} = \sqrt{4 a^{2} + 4 a^{2}} = \sqrt{8 a^{2}} = 2 \sqrt{2} a units B C = \sqrt{{(- \sqrt{3} a + a)}^{2} + {(\sqrt{3} a + a)}^{2}} = a \sqrt{{(1 - \sqrt{3})}^{2} + {(1 + \sqrt{3})}^{2}} = a \sqrt{2 (1 + 3)} = a \sqrt{8} = 2 \sqrt{2} a units A C = \sqrt{{(- \sqrt{3} a - a)}^{2} + {(\sqrt{3} a - a)}^{2}} = a \sqrt{{(\sqrt{3} + 1)}^{2} + {(\sqrt{3} - 1)}^{2}} = a \sqrt{2 (3 + 1)} = a \sqrt{8} = 2 \sqrt{2} a units$
AB = BC = AC
Hence, the points form the vertices of an equilateral triangle.

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