Difference between revisions of "1992 AIME Problems/Problem 13"
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Revision as of 18:24, 4 July 2013
Problem
Triangle 
has 
and 
. What's the largest area that this triangle can have?
Solution
Solution 1
First, consider the triangle in a coordinate system with vertices at 
, 
, and 
. Applying the distance formula, we see that 
.
We want to maximize 
, the height, with 
being the base.
Simplifying gives 
.
To maximize , we want to maximize 
. So if we can write: 
, then 
is the maximum value of (this follows directly from the trivial inequality, because if 
then plugging in 
for 
gives us 
).
.
.
Then the area is 
.
Solution 2
Let the three sides be 
, so the area is 
by Heron's formula. By AM-GM, 
, and the maximum possible area is 
. This occurs when 
.
See also
| 1992 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
