Difference between revisions of "1982 AHSME Problems/Problem 23"
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| − | The | + | == 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 | ||
| + | |||
| + | <math> \textbf{(A)}\ \frac{3}{4}\qquad | ||
| + | \textbf{(B)}\ \frac{7}{10}\qquad | ||
| + | \textbf{(C)}\ \frac{2}{3}\qquad | ||
| + | \textbf{(D)}\ \frac{9}{14}\qquad | ||
| + | \textbf{(E)}\ \text{none of these} </math> | ||
| + | |||
| + | == Solution 1 == | ||
| + | In <math>\triangle ABC,</math> let <math>a=n,b=n+1,c=n+2,</math> and <math>\angle A=\theta</math> for some positive integer <math>n.</math> We are given that <math>\angle C=2\theta,</math> and we wish to find <math>\cos\theta.</math> | ||
| + | |||
| + | |||
| + | |||
| + | == Solution 2 == | ||
| + | |||
| + | == See Also == | ||
| + | {{AHSME box|year=1982|num-b=22|num-a=24}} | ||
| + | {{MAA Notice}} | ||
Revision as of 18:01, 14 September 2021
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
In 
let 
and 
for some positive integer 
We are given that 
and we wish to find 
Solution 2
See Also
| 1982 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 22 |
Followed by Problem 24 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
