If PQRS is a parallelogram, what is the ratio of the area of triangle PQS to…

For a triangle and a parallelogram sharing the same base and height, area of triangle equals half of base times height, and area of parallelogram equals base times height. Also, any diagonal of a parallelogram divides it into two triangles of equal area, each equal to half the parallelogram's total area.

1. Draw diagonal QS in parallelogram PQRS. This diagonal splits PQRS into two triangles, PQS and QRS, that share the same base QS.

2. Since PQ is parallel to SR, and PS is parallel to QR (opposite sides of the parallelogram), both triangles PQS and QRS have the same perpendicular height from the diagonal QS to the parallel sides. Hence the two triangles are equal in area.

3. Because the two triangles together make up the whole parallelogram and are equal in area, each triangle's area equals half of the parallelogram's area: area of PQS equals half of area of PQRS.

4. So, area of triangle PQS is to area of parallelogram PQRS as 1 is to 2.

Cross-check using coordinates: let P = (0, 0), Q = (a, 0), S = (b, c), and R = (a + b, c), so that PQRS is a parallelogram. The area of parallelogram PQRS, using base PQ of length a and height c, is a times c. The area of triangle PQS, using the coordinate formula, is half of the absolute value of (a times c minus b times 0), which is half of a times c. The ratio of the two areas is exactly one half, confirming the ratio 1 : 2.