1982 AHSME Problems/Problem 23
Contents
Problem
The lengths of the sides of a triangle are consecutive integers, and the largest angle is twice the smallest angle. The cosine of the smallest angle is
Solution 1 (Law of Sines and Law of Cosines)
In let
and
for some positive integer
We are given that
and we need
We apply the Law of Cosines to solve for
We apply the Law of Sines to and
respectively:
Recall that
holds for all
We simplify and rearrange to solve for
We equate the expressions for
from which
By substitution, the answer is
~MRENTHUSIASM
Solution 2 (Cosines Only)
This solution uses the same variable definitions as Solution 1 does. Moreover, we conclude that from the second paragraph of Solution 1.
We apply the Law of Cosines to solve for
By the Double-Angle Formula
we have
from which
Recall that
is a positive integer, so
By substitution, the answer is
~MRENTHUSIASM
See Also
1982 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
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