Difference between revisions of "1995 AHSME Problems/Problem 23"
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Latest revision as of 14:05, 5 July 2013
Problem
The sides of a triangle have lengths 
and 
, where is an integer. For how many values of is the triangle obtuse?
Solution
By the Law of Cosines, a triangle is obtuse if the sum of the squares of two of the sides of the triangles is less than the square of the third. The largest angle is either opposite side 
or side . If is the largest side,
![\[15^2 >11^2 + k^2 \Longrightarrow k < \sqrt{104}\]](http://latex.artofproblemsolving.com/f/f/7/ff78bd7dc3274df87c40a1c5d759dd3dae5cf52c.png)
By the Triangle Inequality we also have that 
, so can be 
, or 
values.
If is the largest side,
![\[k^2 >11^2 + 15^2 \Longrightarrow k > \sqrt{346}\]](http://latex.artofproblemsolving.com/3/4/e/34e4fa5e979a889c96a3be9b3f2b02869c064d7d.png)
Combining with the Triangle Inequality 
, or 
values. These total 
values of .
See also
| 1995 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. 
