1998 AJHSME Problems/Problem 20

Problem

Let $PQRS$ be a square piece of paper. $P$ is folded onto $R$ and then $Q$ is folded onto $S$ . The area of the resulting figure is 9 square inches. Find the perimeter of square $PQRS$ .

$[asy] draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); label("$P$",(0,2),SE); label("$Q$",(2,2),SW); label("$R$",(2,0),NW); label("$S$",(0,0),NE); [/asy]$

$\text{(A)}\ 9 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 24 \qquad \text{(E)}\ 36$

Solution

After both folds are completed, the square would become a triangle that has an area of $\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$ of the original square.

Since the area is $9$ square inches for $\frac{1}{4}$ of the square, $9\times4=36$ square inches is the area of square $PQRS$

The length of the side of a square that has an area of $36$ square inches is $\sqrt{36}=6$ inches.

Each side is $6$ inches, so the total perimeter is $6\times4=24=\boxed{D}$

1998 AJHSME (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
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1998 AJHSME Problems/Problem 20

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