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 | ||
