Question

Without using Pythagoras theorem, show that points A (4, 4), B (3, 5) and C (– 1, – 1) are the vertices of a right-angled triangle.

Answer 1

Given, A(4, 4) = (x₁, y₁), B(3, 5) = (x₂, y₂), C (– 1, – 1)  = (x₃, y₃)

Slope of AB = `(y_2 - y_1)/(x_2 - x_1) = (5 - 4)/(3 - 4)` = – 1

Slope of BC = `(y_3 - y_2)/(x_3 - x_2) = (-1 - 5)/(-1 - 3) = (-6)/(-4) = 3/2`

Slope of AC = `(y_3 - y_1)/(x_3 - x_1) = (-1 - 4)/(-1 - 4) = (-5)/(-5)  = 1`

Slope of AB x slope of AC = – 1 x 1 = – 1
∴ side AB ⊥ side AC
∴ ΔABC is a right angled triangle, right angled at A.
∴ The given points are the vertices of a right angled triangle.