CAT 2022 Slot 1 · Quant · Question 1

In a parallelogram ABCD the vertices are named in order, so AB and DC are opposite sides.

Opposite sides of a parallelogram are equal and parallel, which as vectors means \vec{AB}=\vec{DC}.

Writing that out, B-A=C-D.

Rearranging for the unknown vertex, D=A+C-B.

The x-coordinate is 1+\left( -2 \right)-3=-4.

The y-coordinate is 1+8-4=5.

So D is \left( -4,5 \right) — the first option.

As a check, the diagonals of a parallelogram bisect each other: the midpoint of AC is \left( \frac{1-2}{2},\frac{1+8}{2}\right)=\left( -0.5,4.5 \right), and the midpoint of BD is \left( \frac{3-4}{2},\frac{4+5}{2}\right)=\left( -0.5,4.5 \right) — the same point.

Shortcut. C sits 5 units left and 4 units above B, so D must sit 5 units left and 4 units above A. That gives \left( 1-5,\,1+4 \right)=\left( -4,5 \right) by inspection, with no formula at all.