Problem 12 · 2024 Math Kangaroo
Hard
Geometry & Measurement
areapythagorean-triple
A rectangle is split into three pieces of equal area, as shown. One piece is an equilateral triangle with sides of length 4 cm; the other two are trapezoids. How long is the shorter of the two parallel sides of a trapezoid?
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Answer: B — \(\sqrt{3}\)
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Hint 1 of 2
In the picture a full side of the triangle (length 4) lies along the short side of the rectangle, so that side fixes the rectangle's height.
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Hint 2 of 2
Each of the three pieces has the same area, and a trapezoid's area is its average width times its height.
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Approach: use equal areas; the trapezoid's two parallel sides average to a known value
- The triangle's vertical side is 4, so the rectangle is 4 tall, and the triangle's area is \(\frac{\sqrt3}{4}\cdot4^2=4\sqrt3\).
- All three pieces are equal, so the rectangle's area is \(12\sqrt3\) and its width is \(\frac{12\sqrt3}{4}=3\sqrt3\).
- A trapezoid has height 2 (half the rectangle) and longer parallel side \(3\sqrt3\); its area \(4\sqrt3=\tfrac12(3\sqrt3+x)\cdot 2\) gives the shorter side \(x=\sqrt3\).
- So the shorter parallel side is \(\sqrt3\), answer B.
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