1958 AMC 12 Problem 35
Attempt Problem 35 of the 1958 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 1958 AMC 12 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
35.
A triangle is formed by joining three points whose coordinates are integers. If the -unit and the -unit are each inch, then the area of the triangle, in square inches:
must be an integer
may be irrational
must be irrational
must be rational
will be an integer only if the triangle is equilateral
Answer: D
Small Hint:
Translate one vertex to the origin and use the coordinate determinant for twice the area
Big Hint:
The determinant of integer coordinates is an integer
Solution:
After translating one vertex to the origin, write the other two as and with all coordinates integers. The area is Since is an integer, the area is an integer or a half-integer, and in either case it is rational.
Therefore, the correct answer is D.
Problem 35 in Other Years
1950 AMC 12 · 1951 AMC 12 · 1952 AMC 12 · 1953 AMC 12 · 1954 AMC 12 · 1955 AMC 12 · 1956 AMC 12 · 1957 AMC 12 · 1959 AMC 12 · 1960 AMC 12 · 1961 AMC 12 · 1962 AMC 12 · 1963 AMC 12 · 1964 AMC 12 · 1965 AMC 12 · 1966 AMC 12 · 1967 AMC 12 · 1968 AMC 12 · 1969 AMC 12 · 1970 AMC 12 · 1971 AMC 12 · 1972 AMC 12 · 1973 AMC 12