Problem: Let a,b,c and d be the lengths of sides MN,NP,PQ and QM, respectively, of quadrilateral MNPQ. If A is the area of MNPQ, then
Answer Choices:
A. A=(2a+c)(2b+d) if and only if MNPQ is convex
B. A=(2a+c)(2b+d) if and only if MNPQ is a rectangle
C. A⩽(2a+c)(2b+d) if and only if MNPQ is a rectangle
D. A⩽(2a+c)(2b+d) if and only if MNPQ is a parallelogram
E. A⩾(2a+c)(2b+d) if and only if MNPQ is a parallelogram
Solution:
Since A is the sum of the areas of the triangles into which MNPQ is divided by diagonal MP,
A=21absinN+21cdsinQ.
Similarly,
A=21adsinM+21bcsinP.
Therefore
A⩽41(ab+cd+ad+bc)=(2a+c)(2b+d).
The inequality is an equality if and only if sinM=sinN=sinP=sinQ=1, i.e., if and only if MNPQ is a rectangle.