Difference between revisions of "2013 AIME I Problems/Problem 9"
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Proceed with the same labeling as in Solution 1. | Proceed with the same labeling as in Solution 1. | ||
<math>\angle B = \angle C = \angle A = \angle PDQ = 60^\circ</math> | <math>\angle B = \angle C = \angle A = \angle PDQ = 60^\circ</math> | ||
− | <math>\angle PDB + \angle PDQ + \angle QDC = \angle QDC + \angle | + | <math>\angle PDB + \angle PDQ + \angle QDC = \angle QDC + \angle CQD + \angle C = 180^\circ</math> |
Therefore, <math>\angle PDB = \angle DQC</math>. | Therefore, <math>\angle PDB = \angle DQC</math>. | ||
Similarly, <math>\angle BPD = \angle QDC</math>. | Similarly, <math>\angle BPD = \angle QDC</math>. | ||
Line 56: | Line 56: | ||
The solution is <math>39 + 39 + 35 = \boxed{113}</math>. | The solution is <math>39 + 39 + 35 = \boxed{113}</math>. | ||
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== See also == | == See also == | ||
{{AIME box|year=2013|n=I|num-b=8|num-a=10}} | {{AIME box|year=2013|n=I|num-b=8|num-a=10}} |
Revision as of 18:56, 29 March 2013
Contents
Problem 9
A paper equilateral triangle has side length 12. The paper triangle is folded so that vertex
touches a point on side
a distance 9 from point
. The length of the line segment along which the triangle is folded can be written as
, where
,
, and
are positive integers,
and
are relatively prime, and
is not divisible by the square of any prime. Find
.
Solution 1
Let and
be the points on
and
, respectively, where the paper is folded.
Let be the point on
where the folded
touches it.
Let ,
, and
be the lengths
,
, and
, respectively.
We have ,
,
,
,
, and
.
Using the Law of Cosines on :
Using the Law of Cosines on :
Using the Law of Cosines on :
The solution is .
Solution 2
Proceed with the same labeling as in Solution 1.
Therefore,
.
Similarly,
.
Now,
and
are similar triangles.
Solving this system of equations yields
and
.
Using the Law of Cosines on
:
The solution is .
See also
2013 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |