Question

ABCD is a parallelogram where A(x, y), B(5, 8), C(4, 7) and D(2, −4). Find:

1. co-ordinates of A.
2. equation of diagonal BD.

Answer 1

In parallelogram ABCD, A(x, y), B(5, 8), C(4, 7) and D(2, −4).

The diagonals of the parallelogram bisect each other.

O is the point of intersection of AC and BD

Since O is the midpoint of BD, its coordinates will be

`((2 + 5)/2, (-4 + 8)/2)` or `(7/2, 4/2)` or `(7/2, 2)`

i. Since O is the midpoint of AC also,

`(x + 4)/2 = 7/2`

`=>` x + 4 = 7 

`=>` x = 7 – 4 

∴ x = 3

`(y +7)/2 = 2`

`=>` y + 7 = 4

`=>`  y = 4 – 7

∴ y = –3

Thus, Coordinates of A are (3, –3)

ii. `(y - y_1)/(y_2 - y_1) = (x - x_1) /(x_2 - x_1)`

`=> y - y_1 = ((y_2 - y_1))/((x_2 - x_1)) xx (x - x_1)`

`=> y + 4 = (8 + 4)/(5 - 2) xx (x - 2)`

`=> y + 4 = 12/3 xx (x - 2)`

`=>` y + 4 = 4(x – 2)

`=>` y + 4 = 4x – 8

`=>` 4x – y = 12