Problem 27 · 2025 Math Kangaroo
Stretch
Geometry & Measurement
area-fractionratio
In quadrilateral ABCD, the points N and K are marked on sides BC and AD so that BN = 2·NC and AK = KD. The areas of triangles ABN and CKD are shown in the figure. What is the area of quadrilateral ABCD?
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Answer: A — 13
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Hint 1 of 2
A point that splits a side in a ratio splits a triangle's area in the same ratio (same height).
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Hint 2 of 2
Stretch triangle ABN up to ABC using BN:NC, and double CKD up to ACD using the midpoint K.
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Approach: scale each known triangle to a piece of the quadrilateral
- BN = 2·NC means BN:BC = 2:3, so triangle ABC has area 6·(3/2) = 9.
- K is the midpoint of AD, so triangle ACD = 2·(area CKD) = 2·2 = 4.
- Splitting ABCD by diagonal AC: area = \(9 + 4 = 13\), which is (A).
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