Problem 28 · 2020 Math Kangaroo
Stretch
Geometry & Measurement
area-decompositionarea
A large rectangular plot is divided into two lots that are separated from each other by an ABCD fence, as shown in the picture. The AB, BC and CD parts of this fence are parallel to the sides of the rectangle and have lengths of 30 m, 24 m and 10 m, respectively. The owners of these lands have agreed to knock down the fence and make a new straight AE fence, without changing the area of each of the lands. How far from point D should the end E of the fence be?
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Answer: C — 12 m
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Hint 1 of 2
The straight fence must keep each lot’s area, so the area swept on each side cancels.
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Hint 2 of 2
Compare the area to one side of the staircase fence with the area under the straight line.
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Approach: balance areas to locate E on the top edge
- With A at the bottom, the staircase keeps a left area of 24·10 = 240 beyond the step.
- A straight fence A–E to a point E on the top edge gives a left area of 20·(horizontal offset of E).
- Equating, E sits 12 m horizontally from A, i.e. 12 m left of D.
- So E is 12 m from D.
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