Question

Show that points $P(1,–2),Q(5,2),R(3,–1),S(–1,–5)$ are the vertices of a parallelogram

Answer 1

The given points are $P(1,–2),Q(5,2),R(3,–1),S(–1,–5)$
Slope of $PQ=\frac{2+2}{5-1}=\frac{4}{4}=1$
Slope of $QR=\frac{-1-2}{3-5}=\frac{-3}{-2}=\frac{3}{2}$
Slope of $RS=\frac{-5+1}{-1-3}=\frac{-4}{-4}=1$
Slope of $PS=\frac{-5+2}{-1-1}=\frac{-3}{-2}=\frac{3}{2}$
Slope of $PQ=$Slope of $RS$ and Slope of $QR=$ Slope of $PS$
So, $PQ||RS$ and $QR||PS$
Thus, $PQRS$ is a parallelogram.