Difference between revisions of "2002 AIME II Problems/Problem 14"
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== Solution == | == Solution == | ||
− | Let the circle intersect <math>\overline{PM}</math> at <math>B</math>. Then note <math>\triangle OPB</math> and <math>\triangle MPA</math> are similar. Also note that <math>AM = BM</math> by power of a point. So we have | + | Let the circle intersect <math>\overline{PM}</math> at <math>B</math>. Then note <math>\triangle OPB</math> and <math>\triangle MPA</math> are similar. Also note that <math>AM = BM</math> by [[power of a point]]. So we have |
<cmath>\frac{19}{AM} = \frac{152-2AM}{152}</cmath> | <cmath>\frac{19}{AM} = \frac{152-2AM}{152}</cmath> | ||
Solving, <math>AM = 38</math>. So the ratio of the side lengths of the triangles is 2. Therefore, | Solving, <math>AM = 38</math>. So the ratio of the side lengths of the triangles is 2. Therefore, | ||
<cmath>\frac{PB+38}{OP}= 2 \text{ and } \frac{OP+19}{PB} = 2</cmath> | <cmath>\frac{PB+38}{OP}= 2 \text{ and } \frac{OP+19}{PB} = 2</cmath> | ||
− | so | + | so <math>2OP = PB+38</math> and <math>2PB = OP+19.</math> Substituting for <math>PB</math>, we see that <math>4OP-76 = OP+19</math>, so <math>OP = \frac{95}3</math> and the answer is <math>\boxed{098}</math>. |
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− | Substituting for <math>PB</math>, <math>4OP-76 = OP+19</math>, so <math>OP = \frac{95}3</math> and the answer is <math>\boxed{098}</math>. | ||
== See also == | == See also == |
Revision as of 21:40, 25 August 2011
Problem
The perimeter of triangle is
, and the angle
is a right angle. A circle of radius
with center
on
is drawn so that it is tangent to
and
. Given that
where
and
are relatively prime positive integers, find
.
Solution
Let the circle intersect at
. Then note
and
are similar. Also note that
by power of a point. So we have
Solving,
. So the ratio of the side lengths of the triangles is 2. Therefore,
so
and
Substituting for
, we see that
, so
and the answer is
.
See also
2002 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |