Problem:
The three side lengths of a triangle are in arithmetic progression with shortest side of length One of the interior angles measures . What is the area of the triangle?
Answer Choices:
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Solution:
Let and be positive real numbers such that the side lengths are , , and . Then the longest side, that of length , must be opposite the angle. By the Law of Cosines,
Simplifying yields , so and . Therefore the three sides have lengths . Because the shortest side has length 6 , the three side lengths are .\
The area of the triangle is
Alternatively, one can use Heron's Formula with to find the area.
Note: A triple of integers that are the side lengths of a triangle with exactly one angle is called an Eisenstein triple, similar to a Pythagorean triple for a triangle with a angle. Some examples of such triples are and . Similarly, there are other triangles with integer side lengths and a angle, such as and .
The problems on this page are the property of the MAA's American Mathematics Competitions