If two opposite vertices of a square are (1, 7) and (-1, −1), then find the coordinates of its remaining two vertices.
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Step-by-step explanation:
Given If two opposite vertices of a square are (1, 7) and (-1, −1), then find the coordinates of its remaining two vertices.
- So given a square ABCD, so coordinates of A is (1,7) and C is (- 1, -1)
- All sides are equal in a square.
- So AB = BC
- Let coordinates of B be (x,y)
- So we have (x – 1)^2 + (y – 7)^2 = (x + 1)^2 + (y + 1)^2
- So x^2 + 1 – 2x + y^2 + 49 – 14 y = x^2 + 1 + 2x + y^2 + 1 + 2y
- So we get – 4x – 16 y = - 48
- 4x + 16 y = 48
- Or x + 4y = 12 ------------1
- So x = 12 – 4y
- In a square the diagonal is equal to √2 times the side of the square.
- So AC = √2 AB
- So AC^2 = 2 AB^2
- So (- 1 – 1)^2 + (- 1 – 7)^2 = 2 [(x – 1)^2 + (y – 7)^2]
- So (- 2)^2 + (- 8)^2 = 2 {x^2 + 1 – 2x + y^2 + 49 – 14y)
- 4 + 64 = 2(x^2 + y^2 – 2x – 14y + 50)
- 34 = x^2 + y^2 – 2x – 14 y + 50
- Substituting the value of x we get
- 34 = (12 – 4y)^2 + y^2 – 2 (12 – 4y) – 14 y + 50
- 34 = 144 + 16 y^2 – 96 y + y^2 – 24 + 8y – 14y + 50
- 17 y^2 – 102 y + 136 = 0
- Or y^2 – 6y + 8 = 0
- Or y^2 – 4y – 2y + 8 = 0
- Or y(y – 4) – 2(y – 4) = 0
- Or (y – 4) (y – 2) = 0
- Or y = 4 , 2
- For y = 4
- Now x + 4y = 12
- So x + 16 = 12
- Or x = - 4
- For y = 2
- Now x + 4 (2) = 12
- So x + 8 = 12
- Or x = 4
- So coordinates are A(1,7). C(- 1, - 1) B(4, 2) D(- 4, 4)
Reference link will be
https://brainly.in/question/2758279
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