Question

The ends of a diagonal of a parallelogram are $(3,-4)$and $(-6,5)$, If the third vertex is $(-2,1)$, then the fourth is-

• A. $(1,0)$
• B. $(-1,0)$
• C. $(0,1)$
• D. $(0,-1)$

Answer 1

The correct option is B $(-1,0)$

Let $ABCD$ is a parallelogram and $A(3,-4),C(-6,5)$ and $B(-2,1)$

Let fourth vertex be $D(x,y)$

We know that diagonals of a parallelogram bisect each other.

$\therefore$ Co-ordinates of mid-points of $AC=$ Co-ordinates of mid-points of $BD$

$\Rightarrow$ $(\frac{3-6}{2},\frac{-4+5}{2})=(\frac{-2+x}{2},\frac{1+y}{2})$

$\Rightarrow$ $(\frac{-3}{2},\frac{1}{2})=(\frac{-2+x}{2},\frac{1+y}{2})$

$\Rightarrow$ Now, $\frac{-3}{2}=\frac{-2+x}{2}$ and $\frac{1}{2}=\frac{1+y}{2}$

$\Rightarrow$ $-6=-4+2x$ and $2=2+2y$

$\Rightarrow$ $-2=2x$ and $0=2y$

$\therefore$ $x=-1$ and $y=0$

$\therefore$ The fourth vertex $D(-1,0).$