2007 Alabama ARML TST Problems/Problem 15
Contents
[hide]Problem
Let be a point inside isosceles right triangle
such that
,
,
, and
. Find the area of
.
Solution
Solution 1
Let ,
, and
be the reflections of
over sides
,
, and
, respectively. We then have that
,
, and
. This shows that
. I shall now proceed to find
. This is equal to
Note that and
, so
. Similarly,
and
. Now note that
and
. Therefore
and
. Also note that
and
. We also know that
,
, and
are collinear, so
. This shows that
is a 5-12-13 right triangle, so it has area
, so
Solution 2
Rotate the diagram by 90 degrees about so that
goes to
,
goes to a point
, and
goes to
. Since the image of
under this rotation is
,
. Since
is a 45-45-90 right triangle,
. Thus,
is a 5-12-13 right triangle, with
. Note that
, so by Law of Cosines on triangle
,
so .
See also
2007 Alabama ARML TST (Problems) | ||
Preceded by: Problem 14 |
Followed by: Last Problem | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |