Question

The points $A(1,-2),B(2,3),C(k,2)$ and $D(-4,-3)$ are the vertices of a parallelogram . Find the value of $k$.

Answer 1

Opposite sides in parallelogram are parallel.

$\Rightarrow$ Slope of line passing through two points $=\frac{y_2-y_1}{x_2-x_1}$

$\Rightarrow$ Slope of $AD=\frac{-3+2}{-4-1}$

$=\frac{-1}{-5}$

$=\frac{1}{5}$

$\Rightarrow$ Slope of $BC=\frac{3-2}{2-k}=\frac{1}{2-k}$

Slope of parallel lines are equal :

$\Rightarrow \frac{1}{5}=\frac{1}{2-k}$

$2-k=5$

$k=-3$

Hence, the answer is $-3.$