1951 AMC 12 Problem 27
Attempt Problem 27 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).
27.
Through a point inside a triangle, three lines are drawn from the vertices to the opposite sides, forming six triangular sections. Then:
The triangles are similar in opposite pairs
The triangles are congruent in opposite pairs
The triangles are equal in area in opposite pairs
Three similar quadrilaterals are formed
None of the above relations is true
Answer: E
Small hint:
The interior point and the three cevians are arbitrary
Big hint:
Move the point very close to one side to test the claimed opposite-area relation
Solution:
No angle or length condition forces opposite sections to be similar or congruent. Their areas also need not match: placing the interior point very close to one side makes the sections adjoining that side arbitrarily small without forcing their opposite sections to be small. No quadrilaterals are among the six sections.
Thus, the correct answer is E.
Problem 27 in other years
- 1950 AMC 12
- 1952 AMC 12
- 1953 AMC 12
- 1954 AMC 12
- 1955 AMC 12
- 1956 AMC 12
- 1957 AMC 12
- 1958 AMC 12
- 1959 AMC 12
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- 1999 AMC 12