2007 AIME I Problems/Problem 15
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Problem
Let 
be an equilateral triangle, and let 
and 
be points on sides 
and 
, respectively, with 
and 
. Point 
lies on side 
such that angle 
. The area of triangle 
is 
. The two possible values of the length of side are 
, where 
and 
are rational, and 
is an integer not divisible by the square of a prime. Find .
Solution
Denote the length of a side of the triangle 
, and of 
as 
. The area of the entire equilateral triangle is 
. Add up the areas of the triangles using the 
formula (notice that for the three outside triangles, 
): 
. This simplifies to 
. Some terms will cancel out, leaving 
.
is an external angle to 
, from which we find that 
, so 
. Similarly, we find that 
. Thus, 
. Setting up a ratio of sides, we get that 
. Using the previous relationship between and , we can solve for .
Use the quadratic formula, though we only need the root of the discriminant. This is 
. The answer is 
.
See also
| 2007 AIME I (Problems • Resources) | ||
| Preceded by Problem 14 | Followed by Last Question | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
