2009 AIME II Problems/Problem 10
Four lighthouses are located at points A, B, C, and D. The lighthouse at A is 5 kilometers from the lighthouse at B, the lighthouse at B is 12 kilometers from the lighthouse at C, and the lighthouse at A is 13 kilometers from the lighthouse at C. To an observer at A, the angle determined by the lights at B and D and the angle determined by the lights at C and D are equal. To an observer at C, the angle determined by the lights at A and B and the angle determined by the lights at D and B are equal. The number of kilometers from A to D is given by (p*sqrt (q))/r, where p, q, and r are relatively prime positive integers, and r is not divisible by the square of any prime. Find p+q+r.
Solution
Let be the intersection of
and
. By the Angle Bisector Theorem,
/
=
/
, so
=
and
=
, and
+
=
=
, so
=
, and
=
. Let
be the altitude from
to
. It can be seen that triangle
is similar to triangle
, and triangle
is similar to triangle
. If
=
, then
=
,
=
, and
= (
*sqrt(
))*
. Since
+
=
=
,
=
, and
= (
*sqrt (
))/
. The answer is
+
+
=
.
See Also
2009 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |