Question

Find whether the points (b, b), (−b, −b) and (2b, 2b) are the vertices of an equilateral triangle.

Answer 1

Let the points (b, b), (-b, -b) and (2 b, 2 b) represent the vertices A, B, and C of an equilateral triangle respectively.
Distance between the points is given by
$\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$
$AB=\sqrt{{(-b-b)}^2+{(-b-b)}^2}$
$=\sqrt{{(-2b)}^2+{(-2b)}^2}$
$=\sqrt{8(b{)}^2}=2\sqrt{2}b$
$BC=\sqrt{{(2b-(-b\left)\right)}^2+{(2b-(-b\left)\right)}^2}$
$=\sqrt{{\left(3b\right)}^2+{\left(3b\right)}^2}$
$=\sqrt{9(b{)}^2+9(b{)}^2}=\sqrt{18(b{)}^2}=3\sqrt{2}b$
$AC=\sqrt{{(2b-b)}^2+{(2b-b)}^2}$
$=\sqrt{{\left(b\right)}^2+(b{)}^2}=\sqrt{2}b$
$\Rightarrow$ AB ≠ BC
As two sides are not equal in length, ABC is not an equilateral triangle.