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Problem 576

Geometry Difficulty 2.7 Find the answer CEMC Pascal · Canada · 2012

In the diagram, rectangle $PRTV$ is divided into four rectangles. The area of rectangle $PQXW$ is 9. The area of rectangle $QRSX$ is 10. The area of rectangle $XSTU$ is 15.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Let $PQ=a$ , $QR=b$ , $PW=c$ , and $WV=d$ .

Since the large rectangle is divided into four smaller rectangles, then $PQ=WX=VU=a$ , $QR=XS=UT=b$ , $PW=QX=RS=c$ , and $WV=XU=ST=d$ .

Since the area of rectangle $PQXW$ is 9, then $ac=9$ .

Since the area of rectangle $QRSX$ is 10, then $bc=10$ .

Since the area of rectangle $XSTU$ is 15, then $bd=15$ .

The area of rectangle $WXUV$ is $ad$ . We want to determine the value of $ad$ .

If we multiply the equations $ac=9$ and $bd=15$ , we obtain $(ac)(bd)=9\times 15$ , or $abcd=135$ .

We divided this equation by the equation $bc=10$ .

This gives $\dfrac{abcd}{bc} = \dfrac{135}{10} = \dfrac{27}{2}$ , from which $ad=\dfrac{27}{2}$ .

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