Problem 20 · 2023 Math Kangaroo
Stretch
Geometry & Measurement
area-decomposition
A pentagon is cut into smaller parts as shown in the diagram. The numbers in the triangles state the area of the according triangle. How big is the area P of the grey quadrilateral?
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Answer: C — 16
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Hint 1 of 2
Triangles that share the same height have areas in the ratio of their bases.
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Hint 2 of 2
Use those base ratios to chase the unknown areas around the figure until the grey piece is forced.
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Approach: propagate area ratios through shared-height triangles
- The labelled triangle areas fix the ratios in which the diagonals cut each other, via equal-height comparisons.
- Carrying those ratios through the figure determines every sub-area, and hence the grey quadrilateral.
- Its area P comes out to 16.
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