Question

If the points A(2, 2), B(– 2, – 3), C(1, – 3) and D(x, y) form a parallelogram then find the value of x and y.

Answer 1

Let A(2, 2), B(– 2, – 3), C(1, – 3) and D(x, y) are the vertices of a parallelogram.

Slope of a line = `(y_2 - y_1)/(x_2 - x_1)`

Slope of AB = `(-3 - 2)/(-2 - 2) = (-5)/(-4) = 5/4`

Slope of BC = `(-3 + 3)/(-2 - 1) = 0/(-3)` = 0

Slope of CD = `(y + 3)/(x - 1)`

Slope of AD = `(y - 2)/(x - 2)`

Since ABCD is a parallelogram

Slope of AB = Slope of CD

`5/4 = (y + 3)/(x - 1)`

5(x – 1) = 4 (y + 3)

5x – 5 = 4y + 12

5x – 4y = 12 + 5

5x – 4y = 17  ...(1)

Slope of BC = Slope of AD

0 = `(y - 2)/(x - 2)`

y – 2 = 0

y = 2

Substitute the value of y = 2 in (1)

5x – 4(2) = 17

5x – 8 = 17

⇒ 5x = 17 + 8

5x = 25 

⇒ x = `25/5` = 5

The value of x = 5 and y = 2.