Question

The diagonals of the parallelogram PQRS intersect at O. The point A lies on the side SR. Find the ratio of the area of the triangle POQ to that of the triangle PAQ.

Answer 1

OP=OR because the diagonals bisect one another. PQ is a common base for triangles OPQ and APQ. The distance between PQ and RS is the height for all triangles with common base PQ and vertex on RS (the areas of triangles PQS, APQ and PQR are the same). The perpendicular from O to PQ or RS is half the height of the parallelogram because O is the midpoint of PR and QS. So the ratio of the areas of OPQ and APQ is 1:2.