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प्रश्न
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.
उत्तर
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).
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