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

The vertices of the given triangle are (4, 4), (3, 5) and (–1, –1).

The slope (m) of the line non-vertical line passing through the point (x₁, y₁) and

(x₂, y₂) is given by

∴ the slope of the line AB (m₁) =

the slope of the line BC (m₂) =

the slope of the line CA (m₃) =

Now, m₁ m₃ = -1

⇒ Lines AB and CA are perpendicular to each other

∴ given triangle is right-angled at A (4, 4)

And the vertices of the right-angled ∆ are (4, 4) , (3, 5) and (-1, -1)