Question

Given the points A (-3, -1), B (1, 4) C (4, -1) what is the x coordinate of D, if ABCD is a parallelogram?

Answer 1

Given:

ABCD is a parallelogram, A(-3,-1), B(1,4), C(4,-1).

To find:

The x-coordinate of point D.

Solution:

Let the point D be (x,y).

We know that the diagonals of a parallelogram bisect each other. So, third midpoints are same.

Midpoint formula:

`Midpoint=\left((x_1+x_2)/(2),(y_1+y_2)/(2)\right)`

In parallelogram ABCD, AC and BD are two diagonals.

Midpoint of AC = Midpoint of BD

`\left((-3+4)/(2),(-1+(-1))/(2)\right)=\left((1+x)/(2),(4+y)/(2)\right)`

`\left((1)/(2),(-2)/(2)\right)=\left((1+x)/(2),(4+y)/(2)\right)`

`\left((1)/(2),-1\right)=\left((1+x)/(2),(4+y)/(2)\right)`

On comparing both sides, we get

`(1)/(2)=(1+x)/(2)`

`1=1+x`

`1-1=x`

`0=x`

Similarly,

`-1=(4+y)/(2)`

`-2=4+y`

`-2-4=y`

`-6=y`

The coordinates of point D are (0,-6).

Therefore, the x-coordinate of point D is 0.