Problem 21 · 2021 Math Kangaroo
Stretch
Geometry & Measurement
symmetry
An ant climbs from C to A along the path CA and descends from A to B on the stairs, as shown in the diagram. What is the ratio of the lengths of the ascending and descending paths?
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Answer: E — √33
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Hint 1 of 2
The staircase length from A to B is just the total horizontal run plus the total vertical drop of A to B.
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Hint 2 of 2
With the 75° and 60° angles, angle B = 45°; compare CA to (horizontal+vertical) of AB using the law of sines.
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Approach: staircase length = horizontal + vertical; compare with CA
- The descending stairs from A to B have total length equal to the horizontal distance plus the vertical distance between A and B.
- Since the angles at A and C are 75° and 60°, angle B = 45°, so AB rises at 45° and its horizontal+vertical = AB·√2.
- By the law of sines CA = AB·sin45°/sin60° = AB·√2/√3.
- Ratio = CA ÷ (AB·√2) = 1/√3 = √33, choice (E).
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