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Question
Show that the points (a, a), (-a, -a) and `(-asqrt(3), asqrt(3))` are the vertices of an equilateral triangle.
Solution
The given points are let
A(a, a), B(-a, a) and C`(-asqrt(3), asqrt(3))`.
AB = `sqrt((-a -a)^2 + (-a -a)^2)`
= `sqrt(4a^2 + 4a^2) = 2sqrt(2)"a units"`.
BC = `sqrt((-a sqrt(3) + a)^2 + (a sqrt(3) + a)^2)`
= `sqrt(3a^2 + a^2 - 2sqrt(3)a^2 + 3a^2 + a^2 + 2 sqrt(3)a^2)`
= `sqrt(8)a^2 = 2sqrt(2)"a units"`.
and CA = `sqrt((asqrt(3) - a)^2 + (-a sqrt(3) - a)^2)`
= `sqrt(3a^2 + a^2 + 2sqrt(3)a^2 + 3a^2 + a^2 + 2 sqrt(3)a^2)`
= `sqrt(8)a^2 = 2sqrt(2)"a units"`.
as AB = BC = CA = `2sqrt(2)"a"`.
⇒ ΔABC is an equilateral triangle.
Hence proved.
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