Parallelogram ABCD with points P, Q, R, S on AB, BC, CD, and DA respectively. Given AP equals DR (measured along AB from A and along CD from D). If area(ABCD) = 16 cm², determine the area of central quadrilateral PQRS formed by joining P–Q–R–S in order.

Introduction / Context:
When points are chosen symmetrically on the opposite sides of a parallelogram so that matched offsets are equal (here AP = DR), the inner quadrilateral formed by joining the points typically has a fixed fractional area of the whole. This problem tests area-ratio facts that follow from parallel sides and equal-distance strips.

Given Data / Assumptions:

• ABCD is a parallelogram with area = 16 cm².
• P on AB, Q on BC, R on CD, S on DA.
• AP = DR (equal offsets from A on AB and from D on CD).
• Standard interpretation: corresponding offsets on adjacent sides create parallel “strips”.

Concept / Approach:
In a parallelogram, segments drawn at equal distances from opposite vertices along parallel sides carve out congruent triangles at opposite corners. With AP = DR, triangles near A and D clipped by lines through P and R are equal in area; likewise, choosing Q and S on the remaining sides to connect P–Q–R–S yields a central quadrilateral whose area equals exactly one-half of the whole under this equal-offset arrangement.

Step-by-Step Solution:
Equal offsets imply equal “corner triangles” are removed pairwise.The remaining central polygon PQRS occupies the complement of four congruent triangles.Area(PQRS) = 1/2 * Area(ABCD) = 1/2 * 16 = 8 cm².

Verification / Alternative check:
Use coordinates with A(0,0), B(b,0), D(0,h), C(b,h). Take AP = DR = t. Compute corner triangle areas by 1/2 * base * corresponding height; they cancel in opposite pairs, leaving half the total.

Why Other Options Are Wrong:
6, 6.4, 4 cm² contradict the equal-offset symmetry that forces the 1/2 ratio.

Common Pitfalls:
Assuming midpoints are required; here equal offsets (not necessarily midpoints) already guarantee the 1/2 area result.

Final Answer:
8 cm2