1986 AMC 12 Problem 29
Attempt Problem 29 of the 1986 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 1986 AMC 12 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
29.
Two of the altitudes of the scalene triangle have length and If the length of the third altitude is also an integer, what is the biggest it can be?
none of these
Answer: B
Small hint:
For a fixed triangle area, each side is inversely proportional to its corresponding altitude
Big hint:
Apply the triangle inequalities to side lengths proportional to
Solution:
Let the third altitude be Since each side equals twice the common area divided by its altitude, the side lengths are proportional to The two nontrivial triangle inequalities give Thus The largest integral possibility is and its three altitudes are distinct as required for a scalene triangle.
Therefore the correct answer is B.
Problem 29 in other years
- 1950 AMC 12
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