2016 AIME I Problems/Problem 9
Problem
Triangle has
and
. This triangle is inscribed in rectangle
with
on
and
on
. Find the maximum possible area of
.
Solution
Note that if angle is obtuse, it would be impossible for the triangle to inscribed in a rectangle. This can easily be shown by drawing triangle ABC, where
is obtuse. Therefore, angle A is acute. Let angle
and angle
. Then,
and
. Then the area of rectangle
is
. By product-to-sum,
. Since
. The maximum possible value of
is 1, which occurs when
. Thus the maximum possible value of
is
so the maximum possible area of
is
.
-AkashD
See Also
2016 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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