Question

Three vertices of a parallelogram taken in order are $(-1,-6),(2,-5)$ and $(7,2).$ The fourth vertex is :

• A. $(1,4)$
• B. $(1,1)$
• C. $(4,4)$
• D. $(4,1)$

Answer 1

The correct option is A $(1,4)$

Let $A(-1,-6),B(2,-5),C(7,2)$ and $D(x,y)$.

$AC$ and $BD$ are diagonals.

So, we know that the diagonals bisect each other in parallelogram, so the diagonals are divided in ratio $1:1$.

Mid-point co-ordinates of $AC=$ Mid-point co-ordinates of $BD.$

$\Rightarrow$ $(\frac{-1+7}{2},\frac{-6+2}{2})=(\frac{2+x}{2},\frac{-5+y}{2})$

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

Now, comparing co-ordinates

$\Rightarrow$ $3=\frac{2+x}{2}$

$\Rightarrow$ $6=2+x$

$\Rightarrow$ $x=4$

$\Rightarrow$ Now, $-2=\frac{-5+y}{2}$

$\Rightarrow$ $-4=-5+y$

$\Rightarrow$ $y=1$

$\therefore$ The co-ordinates of fourth vertex of parallelogram $D(4,1).$