Question

If the co-ordinates of the vertices of an equilateral triangle with sides of length ‘a’ are (x₁, y₁), (x₂, y₂), (x₃, y₃), then `|(x_1, y_1, 1),(x_2, y_2, 1),(x_3, y_3, 1)|^2 = (3"a"^4)/4`

Answer 1

The area of a triangle with vertices (x₁, y₁), (x₂, y₂) and  (x₃, y₃) is given by

Δ = `1/2 |(x_1, y_1, 1),(x_2, y_2, 1),(x_3, y_3, 1)|`

Also, area of an equilateral triangle with side a is given by

Δ = `sqrt(3)/2 "a"^2`

∴ `1/2 |(x_1, y_1, 1),(x_2, y_2, 1),(x_3, y_3, 1)| = sqrt(3)/4 "a"^2`

Squaring both sides, we get

⇒ Δ² = `1/4 |(x_1, y_1, 1),(x_2, y_2, 1),(x_3, y_3, 1)| = 3/16 "a"^4`

or `|(x_1, y_1, 1),(x_2, y_2, 1),(x_3, y_3, 1)|^2 = (3"a"^4)/4`