Problem 25 · 2010 Math Kangaroo
Stretch
Spatial & Visual Reasoning
tiling-tessellationspatial-reasoning
The picture on the right shows a tile pattern. The side length of the bigger tiles is a and of the smaller ones b. The dotted lines (horizontal and tilted) include an angle of 30°. How big is the ratio a:b?
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Answer: B — \((2+\sqrt{3}):1\)
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Hint 1 of 2
The 30° tilt of the dotted lines is set by how the big and small squares meet.
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Hint 2 of 2
Relate the side lengths through that angle (think of a 15°/75° right triangle).
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Approach: use the 30-degree relation between the tilings
- The dotted lines meet at 30°, which fixes how a small square fits against a big one.
- Working through that geometry, the ratio of the big side to the small side is 2 + √3.
- So a : b = (2 + √3) : 1.
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