2013 Mock AIME I Problems/Problem 12
Contents
[hide]Problem
In acute triangle , the orthocenter
lies on the line connecting the midpoint of segment
to the midpoint of segment
. If
, and the altitude from
has length
, find
.
Solution (easy coordinate bash)
Toss on the coordinate plane with ,
, and
, where
is a real number and
.
Then, the line connecting the midpoints of and
runs from
to
, or more simply the line
.
The orthocenter of will be at the intersection of the altitudes from
and
.
The slope of the altitude from is the negative reciprocal of the slope of
. The slope of
is
, and its negative reciprocal is
. Since the altitude from
passes through the origin, its equation is
.
The altitude from is the vertical line running through
which has equation
.
Thus the lines and
meet on the line
. Substituting the first equation into the second,
.
Multiplying both sides by , we have
.
This rearranges to the quadratic , and completing the square by adding
to each side gives us
. Thus
.
The cases where and
are similar; they merely correspond to two triangles that can each be obtained by reflecting the other across the perpendicular bisector of
, so we consider the case where
.
So .
Thus
The cases where and
are shown below, labeled
and
, respectively, where the dotted line is a midline in both triangles. As you can see, the orthocenter falls perfectly on that line for both triangles, and the value of
is the same for both triangles.
Solution 2 (no coordinates)
Let be the midpoint of
and
be the midpoint of
. Further let
be the foot of the altitude from
,
from
, and
from
, as in the diagram.
Because is a midpoint connector of
amd
is on
and
, we know that
is the midpoint of altitude
. Thus, because, from the problem,
,
. Now we see that
is a midpoint connector of
, so
.
Now, let . We know that
, because they are vertical angles. Because
is right (by the definition of an altitude), we know that
.
is also right, so
.
From , we know that
. From
, we know that
. Equating these two expressions for
, we see that
. From the problem, we know that
.
Now, we can proceed as in Solution 1 by using the quadratic formula to solve for and the Pythagorean Theorem to find
and
. We do this to obtain our answer
.