Problem 7 · 2005 AMC 8
Medium
Geometry & Measurement
pythagorean
Bill walks 12 mile south, then 34 mile east, and finally 12 mile south. How many miles is he, in a direct line, from his starting point?
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Answer: B — 1¼.
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Hint 1 of 2
A wiggly path, but only the net displacement matters. Add up all the south moves into one south distance; the east move stands alone.
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Hint 2 of 2
Net south and net east are perpendicular — the straight-line distance home is the hypotenuse of that right triangle (Pythagoras).
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Approach: collapse the path to net legs, then Pythagoras
- The two south stretches (½ + ½) combine to 1 mile south; the single east stretch is ¾ mile. The direction the moves came in doesn't matter — only the totals.
- Start and finish are corners of a right triangle with legs 1 and ¾. Distance = √(1² + (¾)²) = √(1 + 9/16) = √(25/16) = 5/4 = 1¼.
- Spot the 3-4-5: legs ¾ and 1 are 3 and 4 scaled by ¼, so the hypotenuse is 5 scaled by ¼ = &frac54;. Recognizing the triple skips the square-root work entirely.
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