Problem 13 · 2013 Math Kangaroo
Hard
Geometry & Measurement
area
A and B are opposite vertices of a regular six-sided shape, and the points C and D are the midpoints of two opposite sides. The area of the regular six-sided shape is 60. Determine the product of the lengths of the segments AB and CD.
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Answer: D — 80
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Hint 1 of 2
Write AB (the long diagonal) and CD (the gap between opposite sides) in terms of the hexagon's side.
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Hint 2 of 2
Both the area and the product AB·CD are multiples of side²; take their ratio.
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Approach: express AB, CD, and area via the side length
- For side s: the long diagonal AB = 2s, and the distance between opposite sides CD = s√3.
- So AB · CD = 2√3 · s², while the area is (3√3 / 2) s² = 60.
- Dividing, AB · CD = (4/3) × 60 = 80.
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