Problem 20 · 2022 Math Kangaroo
Hard
Geometry & Measurement
areaarea-decomposition
The diagram shows a square PQRS with side length 1. The point U is the midpoint of the side RS and the point W is the midpoint of the square. The three line segments TW, UW and VW split the square into three equally big areas. How long is the line segment SV?
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Answer: E — \(\tfrac{5}{6}\)
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Hint 1 of 2
Put coordinates on the unit square with W at the centre and U, V as the relevant points.
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Hint 2 of 2
Each of the three regions must have area 1/3; use that to locate V on the top side.
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Approach: coordinates with equal-area condition to find V, then length SV
- Place P, Q, R, S as a unit square with W at the centre and U the midpoint of RS.
- Requiring the three regions cut by TW, UW, VW to each have area 1/3 fixes where V sits along the top side.
- From V's position the segment SV comes out to 5/6.
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