Difference between revisions of "2009 AIME I Problems/Problem 5"
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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, | + | 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 22:36, 20 March 2009
Problem
Triangle has and . Points and are located on and respectively so that , and is the angle bisector of angle . Let be the point of intersection of and , and let be the point on line for which is the midpoint of . If , find .
Solution
Sorry, I fail to get the diagram up here, someone help me.
Since is the midpoint of .
Thus, and the opposite angles are congruent.
Therefore, is congruent to $\bigtriangleupCPK$ (Error compiling LaTeX. Unknown error_msg) because of SAS
angle is congruent to because of CPCTC
That shows is parallel to (also )
That makes triangle congruent to
Thus,
Now let apply angle bisector thm.
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 |