Question

Find the possible co-ordinates of the fourth corner of a parallelogram if its three corners are located at $(3,3),(4,4)$, and $(2,1)$.

• A. $(8,0)$
• B. $(8,-1)$
• C. $(-1,9)$
• D. $(1,0)$
• E. $(4,-3)$

Answer 1

The correct option is D $(1,0)$

Let the corners be $A(3,3);B(4,4);C(2,1)$
Let the fourth vertex D $=(x,y)$
We know that the diagonals of a parallelogram bisect each other. So, the midpoint of AC is same as the midpoint of BD.

Mid point of two points ${(x}_1,y_1)$ and ${(x}_2,y_2)$ is calculated by the formula $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$.
So, midpoint of $AC=$ Mid point of $BD$
$\Rightarrow (\frac{3+2}{2},\frac{3+1}{2})=(\frac{4+x}{2},\frac{4+y}{2})$
$\Rightarrow (\frac{5}{2},\frac{4}{2})=(\frac{4+x}{2},\frac{4+y}{2})$

$\Rightarrow 4+x=5;4+y=4$
$\Rightarrow x=1;y=0$
Hence, $D=(1,0)$.