1995 AIME Problems/Problem 9
Problem
Triangle 
is isosceles, with 
and altitude 
Suppose that there is a point 
on 
with 
and 
Then the perimeter of 
may be written in the form 
where 
and 
are integers. Find 
Solution
Let 
, so 
. Then, 
. Expanding 
using the angle sum identity gives ![\[\tan 3x=\tan(2x+x)=\frac{3\tan x-\tan^3x}{1-3\tan^2x}.\]](http://latex.artofproblemsolving.com/6/1/d/61dd7327d2ec20f5a8c75e6ed0a3b2892b99142a.png)
Thus, 
. Solving, we get 
. Hence, 
and 
by the Pythagorean Theorem. The total perimeter is 
. The answer is thus 
.
See also
| 1995 AIME (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 | ||
