1993 AMC 12 Problem 27
Attempt Problem 27 of the 1993 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1993 AMC 12 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
27.
The sides of have lengths and A circle with center and radius rolls around the inside of always remaining tangent to at least one side of the triangle. When first returns to its original position, through what distance has traveled?
Answer: B
Small hint:
The center traces a smaller triangle whose sides are parallel to the original sides
Big hint:
The original -- triangle has inradius while the center’s path has inradius
Solution:
The center remains one unit from the side it touches, so its path is the triangle formed by the three inward parallel lines at distance This triangle is similar to The original right triangle has area and semiperimeter hence inradius The path triangle has inradius so its linear scale is Its perimeter, and therefore the distance traveled, is Thus the correct answer is B.
Problem 27 in other years
- 1950 AMC 12
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