Problem 25 · 2025 Math Kangaroo
Stretch
Geometry & Measurement
spatial-reasoningsquare-area
In rectangle ABCD, the points E and F lie on side DC (see diagram) so that ∠EBA = ∠DFA = 45° and AB + EF = 20 cm. How long is side BC?
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Answer: D — 10 cm
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Hint 1 of 2
A 45° line rises exactly as much as it runs, so each slanted line shifts sideways by the rectangle's height as it climbs from the bottom to the top.
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Hint 2 of 2
Write where E and F land on the top edge in terms of the width AB and the height BC, then plug into AB + EF = 20.
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Approach: use the 45° lines to locate E and F on the top side
- The 45° line from B reaches the top after moving in by the height \(BC\), and likewise the 45° line from A; so each of E and F sits a distance \(BC\) horizontally inside an end of the top edge.
- Measuring along the top edge then gives \(EF=2\,BC-AB\), so \(AB+EF=2\,BC\); the two slants effectively double the height.
- So \(2\,BC=20\), giving \(BC=\) 10 cm, which is (D).
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