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.
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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.