Difference between revisions of "2000 AIME II Problems/Problem 10"
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== Solution == | == Solution == | ||
− | Call the center of the circle O. By drawing the lines from O tangent to the sides and from O to the vertices of the quadrilateral, eight congruent right triangles are formed. Thus, <math>\angle{AOP}+\angle{POB}+\angle{COQ}+\angle{QOD}=180</math>, or <math>(arctan(19 | + | Call the center of the circle <math>O</math>. By drawing the lines from <math>O</math> tangent to the sides and from <math>O</math> to the vertices of the quadrilateral, eight congruent right triangles are formed. |
+ | |||
+ | Thus, <math>\angle{AOP}+\angle{POB}+\angle{COQ}+\angle{QOD}=180</math>, or <math>(\arctan(\tfrac{19}{r})+\arctan(\tfrac{26}{r}))+(\arctan(\tfrac{37}{r})+\arctan(\tfrac{23}{r}))=180</math>. | ||
+ | |||
+ | Take the <math>\tan</math> of both sides and use the identity for <math>\tan(A+B)</math> to get <math>\tan(\arctan(\tfrac{19}{r})+\arctan(\tfrac{26}{r}))+\tan(\arctan(\tfrac{37}{r})+\arctan(\tfrac{23}{r}))=n\cdot0=0</math>. | ||
+ | |||
+ | Use the identity for <math>\tan(A+B)</math> again to get <math>\frac{\tfrac{45}{r}}{1-19\cdot\tfrac{26}{r^2}+\frac{\tfrac{60}{r}}{1-37\cdot\tfrac{23}{r^2}}}=0</math>. | ||
+ | |||
+ | Solving gives <math>r^2=\boxed{647}</math>. | ||
== See also == | == See also == | ||
{{AIME box|year=2000|n=II|num-b=9|num-a=11}} | {{AIME box|year=2000|n=II|num-b=9|num-a=11}} |
Revision as of 03:27, 15 March 2008
Problem
A circle is inscribed in quadrilateral , tangent to
at
and to
at
. Given that
,
,
, and
, find the square of the radius of the circle.
Solution
Call the center of the circle . By drawing the lines from
tangent to the sides and from
to the vertices of the quadrilateral, eight congruent right triangles are formed.
Thus, , or
.
Take the of both sides and use the identity for
to get
.
Use the identity for again to get
.
Solving gives .
See also
2000 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |