Difference between revisions of "1992 AIME Problems/Problem 13"
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== Problem == | == Problem == | ||
− | Triangle <math>ABC</math> has <math>AB=9 | + | Triangle <math>ABC</math> has <math>AB=9</math> and <math>BC: AC=40: 41</math>. What's the largest area that this triangle can have? |
== Solution == | == Solution == |
Revision as of 02:55, 15 March 2008
Problem
Triangle has
and
. What's the largest area that this triangle can have?
Solution
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 for
. This follows directly from the trivial inequality, because if
then plugging in
for
gives us
. So we can keep increasing the left hand side of our earlier equation until
. We can factor
into
. We find
, and plug into
. Thus, the area is
.
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 |