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Geometry Difficulty 3.6 AMC 10/12 Find the answer Canada

A square is cut along a diagonal and reassembled to form the parallelogram $PQRS$ as shown in the diagram.

If $\text{PR=90 mm}$ , what is the area of the original square, in $\text{mm 2}$ ?

Pick one

Solution

Suppose that the original square had side length $x$ mm.

We extend $PQ$ and draw a line through $R$ perpendicular to $PQ$ , meeting $PQ$ extended at $T$ .

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$SRTQ$ is a square, since it has three right angles at $S$ , $Q$ , $T$ (which makes it a rectangle) and since $SR=SQ$ (which makes the rectangle a square).

Now $RT=SQ = x$ mm and $PT=PQ+QT=2x$ mm.

By the Pythagorean Theorem, $PR^2 = PT^2 + RT^2$ and so $90^2 = x^2 + (2x)^2$ .

Therefore, $5x^2 = 8100$ or $x^2= 1620$ .

The area of the original square is $\text{x 2 mm 2}$ , which equals $\text{1620 mm 2}$ .

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.