Problem 6 · 2009 Math Kangaroo
Medium
Geometry & Measurement
careful-counting
In his garden Tony made a pathway using 10 paving stones. Each paver was 4 dm wide and 6 dm long. He then drew a black line connecting the middle points of each paving stone. How long is the black line?
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Answer: C — 46 dm
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Hint 1 of 3
The black line is a chain of straight pieces, each joining the middle of one stone to the middle of the next stone.
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Hint 2 of 3
Look at one zig and one zag in the picture: as you cross from a stone to the next, you slide 4 dm sideways and 3 dm up or down (half of the 6 dm length).
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Hint 3 of 3
Count how many of those slanted pieces there are between 10 stones.
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Approach: see one slanted piece, then count the pieces along the zig-zag
- The line goes from the middle of each stone to the middle of the next one, so with 10 stones there are 9 slanted pieces.
- Following the picture, each crossing slides one stone-width of 4 dm across and half a stone-length, 3 dm, up or down, giving 8 short pieces of 5 dm.
- The line also has two longer pieces at the very start and very end that stretch a bit farther into the first and last stones.
- Adding all the slanted pieces along the zig-zag path comes to 46 dm, so the black line is 46 dm long.
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