Problem 12 · 2023 Math Kangaroo
Hard
Geometry & Measurement
areaarea-decomposition
The diagram shows three adjacent squares with side lengths 3 cm, 5 cm and 8 cm. How big is the area of the shaded trapezium?
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Answer: B — \(\frac{55}{4}\) cm\(^2\)
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Hint 1 of 2
The diagonal runs from the bottom-left corner to the top-right corner; find its slope.
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Hint 2 of 2
The shaded piece sits over the middle (5 cm) square; find the diagonal's height at that square's two edges.
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Approach: use the diagonal's slope to size a trapezoid
- The diagonal goes from (0,0) to (16,8), so it has slope 1/2.
- The middle square spans x = 3 to x = 8; there the diagonal is at heights 1.5 and 4.
- The shaded trapezium has parallel sides 1.5 and 4 and width 5: area = (1.5+4)/2 · 5 = 55/4.
- So the shaded area is 55/4 cm².
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