In parallelogram PQRS, QR is produced to M such that |QR| = |RM|. What fraction of the area of PQMS is the area of PRMS?

Analyze the question: PQRS is a parallelogram and QR is extended to M so that |RM| = |QR|. We need the area of PRMS as a fraction of the area of PQMS.

Concept: In a parallelogram, QR is parallel to PS and |QR| = |PS|. Since M lies on line QR extended, PS is parallel to QM. So PQMS is a trapezium with parallel sides PS and QM.

Solution:

Let |QR| = |PS| = a. Then |QM| = |QR| + |RM| = 2a. Let the perpendicular distance between the parallel lines be h.

• Area of PQMS (trapezium) = ½(PS + QM) × h = ½(a + 2a) × h = (3/2)ah
• PRMS has parallel sides PS and RM, both equal to a, so it is also a trapezium (in fact a parallelogram, since PS ∥ RM and PS = RM). Its area = ½(a + a) × h = ah

Fraction = ah ÷ (3/2)ah = 2/3

Check by triangles: Triangle PQR has area ½ah, and PQMS = PQR + PRMS = ½ah + ah = (3/2)ah. So PRMS is ah / (3/2)ah = 2/3 of PQMS. ✔

Answer: The fraction is 2/3, which is option C.

Tip: When a side is extended by its own length, draw the figure and split it into triangles and parallelograms that share the same height. Then you only need to compare bases.