Difference between revisions of "Mock Geometry AIME 2011 Problems/Problem 3"

m
m (Solution)
Line 1: Line 1:
 
==Problem==  
 
==Problem==  
 
In triangle <math>ABC,</math> <math>BC=9.</math> Points <math>P</math> and <math>Q</math> are located on <math>BC</math> such that <math>BP=PQ=2,</math> <math>QC=5.</math> The circumcircle of <math>APQ</math> cuts <math>AB,AC</math> at <math>D,E</math> respectively. If <math>BD=CE,</math> then the ratio <math>\frac{AB}{AC}</math> can be expressed in the form <math>\frac{m}{n},</math> where <math>m,n</math> are relatively prime positive integers. Find <math>m+n.</math>  
 
In triangle <math>ABC,</math> <math>BC=9.</math> Points <math>P</math> and <math>Q</math> are located on <math>BC</math> such that <math>BP=PQ=2,</math> <math>QC=5.</math> The circumcircle of <math>APQ</math> cuts <math>AB,AC</math> at <math>D,E</math> respectively. If <math>BD=CE,</math> then the ratio <math>\frac{AB}{AC}</math> can be expressed in the form <math>\frac{m}{n},</math> where <math>m,n</math> are relatively prime positive integers. Find <math>m+n.</math>  
 
==Solution==
 
  
 
==Solution==
 
==Solution==

Revision as of 21:05, 1 January 2012

Problem

In triangle $ABC,$ $BC=9.$ Points $P$ and $Q$ are located on $BC$ such that $BP=PQ=2,$ $QC=5.$ The circumcircle of $APQ$ cuts $AB,AC$ at $D,E$ respectively. If $BD=CE,$ then the ratio $\frac{AB}{AC}$ can be expressed in the form $\frac{m}{n},$ where $m,n$ are relatively prime positive integers. Find $m+n.$

Solution

[asy] unitsize(1cm); draw((-1,0)--(8,0)--(0,4)--cycle); draw((0,4)--(1,0)); draw((0,4)--(3,0)); label("$A$",(0,4),N); label("$B$",(-1,0),WSW); label("$C$",(8,0),ESE); label("$P$",(1,0),S); label("$Q$",(3,0),S); draw(circumcircle((0,4),(1,0),(3,0)));  label("$D$",(-0.6,2),SW); label("$E$",(4.5,1.7),NE); [/asy]

By the Power of a Point Theorem on $B$, we have $BD*BA=BP*BQ=2*4=8$. By the Power of a Point on $C$, we have $CE*CA=CQ*CP=5*7=35$. Dividing these two results yields $\frac{BD*BA}{CE*CA}=\frac{8}{35}$. We are given $BD=CE$ and so $\frac{BD}{CE}=1$. Then the previous equation simplifies to $\frac{AB}{AC}=\frac{8}{35}$. Hence $m+n=8+35=\boxed{043}$