Problem 26 · 2010 Math Kangaroo
Stretch
Geometry & Measurement
arithmetic-sequencesymmetry
In the figure, \(\alpha = 7^\circ\). All the lines OA1, A1A2, A2A3, … are equally long. What is the maximum number of lines that can be drawn in this way if no two lines are allowed to intersect each other?
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Answer: D — 13
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Hint 1 of 2
Each equal-length step turns the direction by the apex angle of 7 degrees.
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Hint 2 of 2
Lines can keep being added while the built-up angle stays below 90 degrees.
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Approach: accumulate 7 degrees until it reaches 90
- Because the segments are equal, each new one increases the running angle by 7 degrees.
- Drawing can continue while the total stays under 90: 12x7 = 84 works, 13x7 = 91 does not.
- Counting the lines that fit gives 13.
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