Problem 9 · 2025 Math Kangaroo
Medium
Geometry & Measurement
area-decompositionarea
ABCD is a rectangle (see diagram). The area of the quadrilateral ABPD is 4 cm² and \(DP = \tfrac{1}{3}\,CD\). What is the area of the rectangle ABCD?
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Answer: C — 6 cm²
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Hint 1 of 2
ABPD is a trapezoid with parallel sides AB and DP; relate its area to the whole rectangle.
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Hint 2 of 2
Its area is ½(AB + DP)·AD; with DP = AB/3 this is two-thirds of the rectangle.
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Approach: trapezoid area as a fraction of the rectangle
- ABPD is a trapezoid with parallel sides AB and DP = AB/3 and height AD.
- Its area = ½(AB + AB/3)·AD = (2/3)·(AB·AD) = (2/3)·[ABCD].
- So 4 = (2/3)[ABCD], giving [ABCD] = 6 cm².
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