Problem 24 · 2022 Math Kangaroo
Stretch
Geometry & Measurement
proportion
Two rectangles are inscribed into a triangle as shown in the diagram. The dimensions of the rectangles are \(1\times 5\) and \(2\times 3\) respectively. How big is the height of the triangle in A?
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Answer: B — \(\tfrac{7}{2}\)
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Hint 1 of 3
A corner of each rectangle sits on a slanted side, so the little triangle above each rectangle is similar to the whole triangle.
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Hint 2 of 3
Width-of-rectangle to base behaves like remaining-height to total height — write that proportion for both rectangles.
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Hint 3 of 3
Two such proportions in the unknown base and height let you eliminate the base and solve for the height.
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Approach: each inscribed rectangle cuts off a triangle similar to the whole, giving two proportions
- Let the triangle have base \(b\) and height \(H\) (the height at \(A\)); a rectangle of height \(h\) and width \(w\) inscribed against the base satisfies \(\dfrac{w}{b}=\dfrac{H-h}{H}\) by similar triangles.
- The two rectangles give \(\dfrac{5}{b}=\dfrac{H-1}{H}\) and \(\dfrac{3}{b}=\dfrac{H-2}{H}\) (using the 1\(\times\)5 and 2\(\times\)3 pieces).
- Dividing the two equations removes \(b\): \(\dfrac{5}{3}=\dfrac{H-1}{H-2}\), so \(5(H-2)=3(H-1)\) and \(2H=7\).
- Hence the height is \(H=\tfrac{7}{2}\), choice B.
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