Question

Prove that the points A(–5, 4); B(–1, –2) and C(5, 2) are the vertices of an isosceles right-angled triangle. Find the co-ordinates of D so that ABCD is a square.

Answer 1

We have:

`AB = sqrt((-1 + 5)^2 + (-2 - 4)^2)`

= `sqrt(16 + 36)`

= `sqrt(52)`

`BC = sqrt((-1 + 5)^2 + (-2 - 2)^2)`

= `sqrt(36 + 16)`

= `sqrt(52)`

`AC = sqrt((5 + 5)^2 + (2 - 4)^2)`

= `sqrt(100 + 4)`

= `sqrt(104)`

AB² + BC² = 52 + 52 = 104

AC² = 104

∵ AB = BC and AB² + BC² = AC²

∵ ABC is an isosceles right-angled triangle.

Let the coordinates of D be (x, y).

If ABCD is a square, then,

Mid-point of AC = Mid-point of BD

`((-5 + 5)/(2),(4 + 2)/(2)) = ((x - 1)/(2),(y - 2)/(2))`

`0 = (x - 1)/2, 3 = (y - 2)/2`

x = 1, y = 8

Thus, the co-ordinates of point D are (1, 8).