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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.
Solution
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)
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