Question

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

Answer 1

AB = `sqrt ((-1 +5)^2 (-2-4)^2) = sqrt (16+36) = sqrt 52` units

BC = `sqrt ((-1 - 5)^2 + (-2-2)^2) = sqrt (36 + 36) = sqrt 52` units

AC = `sqrt ((5 + 5)^2 + (2 - 4)^2) = sqrt (100 + 4) = sqrt 104` units

AB² + BC² = 52 + 52 = 104

AC² = 104

∵ AB = AC and AB² + BC² = AC²

∴ ABC  is an isosceles right angled triangle.

Let the coordinates of D be (x , y)

If ABCD is a square ,

Midpoint of AC = mid point of BD

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

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

x = 1 , y = 8

Coordinates of D are (1 , 8)