2011 AIME I Problems/Problem 4
Problem 4
In triangle ,
,
and
. The angle bisector of angle
intersects
at point
, and the angle bisector of angle
intersects
at point
. Let
and
be the feet of the perpendiculars from
to
and
, respectively. Find
.
Solution
Extend such that it intersects lines
and
at points
and
, respectively. Notice that
is a midline; it then follows that
and
. Now notice that
and
are both isosceles. Thus,
and
. Since
is a midline,
. We want to find
, which is just
.
Substituting, the answer is .
See also
2011 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |