1951 AMC 12 Problem 29
Attempt Problem 29 of the 1951 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 1951 AMC 12 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
29.
Of the following sets of data, the only one that does not determine the shape of a triangle is:
The ratio of two sides and the included angle
The ratios of the three altitudes
The ratios of the three medians
The ratio of the altitude to the corresponding base
Two angles
Answer: D
Small Hint:
Determining shape means determining the triangle up to similarity
Big Hint:
A single altitude-to-base ratio fixes an area ratio but can occur in differently shaped triangles
Solution:
Choices A and E determine the angles up to similarity. Ratios of all three altitudes determine reciprocal side ratios, and ratios of all three medians determine side ratios. But a single ratio does not determine the remaining side lengths or angles; many non-similar triangles can share it.
Thus, the correct answer is D.
Problem 29 in Other Years
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