Problem 16 · 2011 Math Kangaroo
Hard
Geometry & Measurement
areaarea-decomposition
There are three horizontal lines and three parallel sloped lines. Both circles shown touch four of the lines. X, Y and Z are the areas of the grey regions, and D is the area of the parallelogram PQRS. At least how many of the areas X, Y, Z and D must you know in order to determine the area of the parallelogram T?
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Answer: A — 1
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Hint 1 of 2
The equally spaced parallel lines split PQRS into pieces with fixed area ratios.
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Hint 2 of 2
Because those ratios are fixed, one known area pins down all the rest, including T.
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Approach: exploit the fixed area ratios from equally spaced lines
- The three-by-three set of parallel lines cuts the parallelogram into regions whose areas stay in fixed proportion.
- Knowing any single one of X, Y, Z or D therefore determines every region, T included.
- So you need just 1 of them.
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