Question

Show that the following points 𝑨(𝒂, 𝟎), 𝑩(−𝒂, 𝟎), and 𝑪(𝟎, 𝒂√𝟑) form an equilateral triangle.

Answer 1

Given: A(a, 0), B(−a, 0), and C(0, a√3) .
Find the distance AB, BC, and CA using the distance formula.
$AB=\sqrt{(}(x_2-x_1{)}^2+(x_2-x_1{)}^2)$
$AB=\sqrt{(}(a-(-a){)}^2+(0-0{)}^2)$
$AB=\sqrt{4}{a}^2=2a$
$BC=\sqrt{(}{x}_2-x_1{)}^2+(x_2-x_1{)}^2$
$BC=\sqrt{(}0-(-a){)}^2+\left(a\sqrt{(}3\right)-0{)}^2$
$BC=\sqrt{(}a2+3a^2)=2a$
$CA=\sqrt{(}{x}_2-x_1{)}^2+(x_2-x_1{)}^2$
$CA=\sqrt{(}0-\left(a\right){)}^2+\left(a\sqrt{(}3\right)-0{)}^2$
$CA=\sqrt{(}a2+3a2)=2a$
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Since AB = BC = CA, ABC is an equilateral triangle.