Difference between revisions of "2009 AIME II Problems/Problem 10"

(New page: Let O be the intersection of BC and AD. By the Angle Bisector Theorem,5/BO = 13/CO, so BO = 5x and CO = 13x, and BO + OC = BC = 12, so x = 2/3, and OC = 26/3. Let P be the altitude fro...)
 
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Let O be the intersection of BC and AD. By the [[Angle Bisector Theorem]],5/BO = 13/CO, so BO = 5x and CO = 13x, and BO + OC = BC = 12, so x = 2/3, and OC = 26/3. Let P be the altitude from D to OC. It can be seen that triangle DOP is similar to triangle AOB, and triangle DPC is similar to triangle ABC. If DP = 15y, then CP = 36y, OP = 10y, and OD = (5*sqrt(13))*y. Since OP + CP = 46y = 26/3, y = 13/69, and AD = (60*sqrt (13))/23. The answer is 60 + 13 + 23 = 096.
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Four lighthouses are located at points A, B, C, and D. The lighthouse A is 5 km away from the lighthouse at B, the lighthouse at B is 12 km from the lighthouse at C, and the lighthouse at A is 13 km 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 are equal, and the angle determined by the lights at D and B are equal. The number of km 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.

Revision as of 17:35, 17 April 2009

Four lighthouses are located at points A, B, C, and D. The lighthouse A is 5 km away from the lighthouse at B, the lighthouse at B is 12 km from the lighthouse at C, and the lighthouse at A is 13 km 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 are equal, and the angle determined by the lights at D and B are equal. The number of km 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.