Question

ABCD is a square . If the coordinates of A and C are (5 , 4) and (-1 , 6) ; find the coordinates of B and D.

Answer 1

Given ABCD is a square.

∴ AB = BC  (all sides of a square are equal)

`sqrt (("x" - 5)^2 + ("y" - 4)^2) = sqrt (("x" + 1)^2 + ("y" - 6)^2)`

squaring both sides ,

x² + 25 - 10x + y² + 16 - 8y = x² + 1 + 2x + y² + 36 - 12y

⇒ - 12 x + 4y + 4 = 0

⇒ - 3x + y + 1 =0

y = 3x - 1   ......(1)

Also , each angle in a square measures 90°

By pythogoras theorem ,

AB² + BC² = AC²

⇒ (5 - x)² + (4 - y)² + (x + 1)² + (y - 6)² = 36 +4

⇒ 25 + x² - 10x + 16 + y² - 8y + x² + 1 + 2x + y² + 36 - 12y

⇒ 2x² + 2y² - 8x - 20y + 38 = 0

⇒ x² + y² - 4x - 10y + 19 = 0

⇒ x² + (3x - 1)² - 4x - 10(3x - 1)+ 19 = 0

⇒ x² + 9x² + 1 - 6x - 4x - 30x + 10  10 = 0

⇒ 10x² - 40x + 30 = 0

⇒ x² - 4x + 3 = 0

⇒ x² - 3x - x + 3 = 0

⇒ x (x - 3) -1 (x - 3) = 0

⇒ (x - 1)(x - 3) = 0

x = 1 ,3

When , x = 1 ,  y = 3(1)-1 = 2       (1 , 2)
            x = 3 , y = 3(3) - 1 = 8      (3 , 8)

Thus , coordinates of B and D are (1 , 2) and (3 , 8)