In square $P Q R S, M$ is the midpoint of $P S$ and $N$ is the midpoint of $S R$. If the area of $\triangle S M N$ is 18 , then the area of $\triangle Q M N$ is
(A) 36
(B) 72
(C) 90
(D) 48
(E) 54

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In square $P Q R S, P S=S R$ and since $M$ and $N$ are midpoints of these sides having equal length, then $M S=S N$.

The area of $\triangle S M N$ is $\frac{1}{2} \times M S \times S N$.

Since this area equals 18 , then $\frac{1}{2} \times M S \times S N=18$ or $M S \times S N=36$ and so $M S=S N=6$ (since they are equal in length).

The side of the square, $P S$, is equal in length to $P M+M S=6+6=12$ (since $M$ is the midpoint of $P S$ ) and so $P S=S R=R Q=Q P=12$.

The area of $\triangle Q M N$ is equal to the area of square $P Q R S$ minus the combined areas of the three right-angled triangles, $\triangle S M N, \triangle N R Q$ and $\triangle Q P M$.

Square $P Q R S$ has area $P S \times S R=12 \times 12=144$.

$\triangle S M N$ has area 18 , as was given in the question.

$\triangle N R Q$ has area $\frac{1}{2} \times Q R \times R N=\frac{1}{2} \times 12 \times 6=36$ (since $S N=R N=6$ ).

$\triangle Q P M$ has area $\frac{1}{2} \times Q P \times P M=\frac{1}{2} \times 12 \times 6=36$.

Thus the area of $\triangle Q M N$ is $144-18-36-36=54$.

ANSWER: (E)