Question

If $A(-2,1),B(a,b),C(4,6)$ and $D(1,2)$ are the vertices of parallelogram. Find $a+b$

Answer 1

In parallelogram the diagonals bisector each other

Midpoint of $DB=$ midpoint of $AC$

$(\frac{-2+4}{2},\frac{1+6}{2})\equiv (\frac{1+9}{2},\frac{2+b}{2})$

(as mid point of $(x_1,y_1)$ and $(x_2,y_2)\ge (\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

$\Rightarrow 1+a=2$

$a=1$

$\Rightarrow 2+b=7$

$b=5$

$\Rightarrow (a,b)=(1,5)$