1995 AIME Problems/Problem 9
Contents
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 1
Let , so
. Then,
. Expanding
using the angle sum identity gives
Thus,
. Solving, we get
. Hence,
and
by the Pythagorean Theorem. The total perimeter is
. The answer is thus
.
Solution 2
In a similar fashion, we encode the angles as complex numbers, so if , then
and
. So we need only find
such that
. This will happen when
, which simplifies to
. Therefore,
. By the Pythagorean Theorem,
, so the perimeter is
, giving us our answer,
.
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 |