Question

If $P(1,2),Q(4,6),R(5,7)$ and $S(a,b)$ are the vertices of the parallelogram $PQRS$, then find $a+b$.

Answer 1

Here, $PQRS$ is a parallelogram. Then,

Midpoint of $PR=$ Mid-point of $QS$

$\Rightarrow$ $(\frac{1+5}{2},\frac{2+7}{2})=(\frac{4+a}{2},\frac{6+b}{2})$

$\Rightarrow$ $(\frac{6}{2},\frac{9}{2})=\frac{4+a}{2},\frac{6+b}{2}$

Now,

$\Rightarrow$ $\frac{4+a}{2}=\frac{6}{2}$

$\Rightarrow$ $4+a=6$

$\therefore$ $a=2$

$\Rightarrow$ $\frac{6+b}{2}=\frac{9}{2}$

$\Rightarrow$ $6+b=9$

$\therefore$ $b=3$

$\therefore$ $S(a,b)=S(2,3)$

$\therefore$ $a+b=5$