Problem 23 · 2020 Math Kangaroo
Stretch
Geometry & Measurement
area-decompositionpythagorean-triple
A rectangular garden was 50 m long and 40 m wide. An artificial lake was built next to it so that the whole arrangement forms a 60 m square. Then a fence was stretched in a straight line, splitting both the garden and the lake into two parts of equal area, as shown. How long is this fence?
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Answer: B — \(30\sqrt{5}\) m
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Hint 1 of 2
A line that halves both the garden and the lake must pass through both their centres.
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Hint 2 of 2
Find the two centres, then measure the segment across the 60×60 square.
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Approach: the bisecting line joins the two region centroids
- A single straight cut that halves both regions must pass through the centroid of the garden (50×40) and the centroid of the L-shaped lake.
- Placing the 60×60 square with corner at the origin, those centroids are at (25, 20) and (36.25, 42.5), so the line has slope 2.
- Crossing the square, that line runs from (15, 0) to (45, 60), a length of \(\sqrt{30^2 + 60^2} = 30\sqrt{5}\) m, option B.
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