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Difference between revisions of "2009 AIME I Problems/Problem 5"

(→‎Solution)
(→‎Solution)
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Sorry, I fail to get the diagram up here, someone help me.
 
Sorry, I fail to get the diagram up here, someone help me.
  
−
Since <math>K</math> is the midpoint of <math>PM, AC</math>.
+
Since <math>K</math> is the midpoint of <math>\overline{PM}, \overline{AC}</math>.
  
 
Thus, <math>AK=CK,PK=MK</math> and the opposite angles are congruent.
 
Thus, <math>AK=CK,PK=MK</math> and the opposite angles are congruent.
  
−
Therefore, triangle <math>AMK</math> is congruent to triangle <math>CPK</math> because of SAS
+
Therefore, <math>\bigtriangleup{AMK}</math> is congruent to <math>\bigtriangleupCPK</math> because of SAS
  
 
angle <math>KMA</math> is congruent to <math>KPA</math> because of CPCTC
 
angle <math>KMA</math> is congruent to <math>KPA</math> because of CPCTC
  
−
That shows <math>AM</math> is parallel to <math>CP</math> (also <math>CL</math>)
+
That shows <math>\overline{AM}</math> is parallel to <math>\overline{CP}</math> (also <math>CL</math>)
  
−
That makes triangle <math>AMB</math> congruent to <math>LPB</math>
+
That makes triangle <math>\bigtriangleup{AMB}</math> congruent to <math>\bigtriangleup{LPB}</math>
  
 
Thus, <math>\frac {AM}{LP}=\frac {AB}{LB}</math>
 
Thus, <math>\frac {AM}{LP}=\frac {AB}{LB}</math>

Revision as of 23:36, 20 March 2009

Problem

Triangle $ABC$ has $AC = 450$ and $BC = 300$. Points $K$ and $L$ are located on $\overline{AC}$ and $\overline{AB}$ respectively so that $AK = CK$, and $\overline{CL}$ is the angle bisector of angle $C$. Let $P$ be the point of intersection of $\overline{BK}$ and $\overline{CL}$, and let $M$ be the point on line $BK$ for which $K$ is the midpoint of $\overline{PM}$. If $AM = 180$, find $LP$.

Solution

Sorry, I fail to get the diagram up here, someone help me.

Since $K$ is the midpoint of $\overline{PM}, \overline{AC}$.

Thus, $AK=CK,PK=MK$ and the opposite angles are congruent.

Therefore, $\bigtriangleup{AMK}$ is congruent to $\bigtriangleupCPK$ (Error compiling LaTeX. Unknown error_msg) because of SAS

angle $KMA$ is congruent to $KPA$ because of CPCTC

That shows $\overline{AM}$ is parallel to $\overline{CP}$ (also $CL$)

That makes triangle $\bigtriangleup{AMB}$ congruent to $\bigtriangleup{LPB}$

Thus, $\frac {AM}{LP}=\frac {AB}{LB}$

\[\frac {AM}{LP}=\frac {AB}{LB}=\frac {AL+LB}{LB}=\frac {AL}{LB}+1\]

Now let apply angle bisector thm.

\[\frac {AL}{LB}=\frac {AC}{BC}=\frac {450}{300}=\frac {3}{2}\]

\[\frac {AM}{LP}=\frac {AL}{LB}+1=\frac {5}{2}\]

\[\frac {180}{LP}=\frac {5}{2}\]

\[LP=\boxed {072}\]

See also

2009 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 4 		Followed by
Problem 6
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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