Problem 16 · 2012 Math Kangaroo
Stretch
Geometry & Measurement
area-decompositionpythagorean-triple
In the diagram we see a rose bed. White roses are growing in the squares that are equally big, red ones are in the big square and yellow ones in the right-angled triangle. The bed has width and height 16 m. How big is the area of the bed?
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Answer: C — 144 m²
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Hint 1 of 2
The triangle is an isosceles right triangle (its two leg-squares are equal), with the big red square sitting on its hypotenuse.
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Hint 2 of 2
Let the red square have side \(d\); then express the whole figure's width and height in terms of \(d\) and use that both equal 16.
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Approach: express everything through the hypotenuse-square side d
- Let the red square (on the hypotenuse) have side \(d\). The right angle sits at the top apex, so the two equal legs have length \(\frac{d}{\sqrt{2}}\), and each white square has side \(\frac{d}{\sqrt{2}}\).
- The two white squares fan out left and right, making the figure \(2d\) wide; stacked on the red square they reach height \(2d\). So \(2d = 16\), giving \(d = 8\).
- Add the pieces: red square \(d^2 = 64\), two white squares \(2\cdot\frac{d^2}{2} = 64\), and the yellow triangle \(\frac{1}{2}\left(\frac{d}{\sqrt2}\right)^2 = \frac{d^2}{4} = 16\).
- Total \(= 64 + 64 + 16 = 144\;\text{m}^2\), choice C.
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