2001 AIME I Problems/Problem 13
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Problem
In a certain circle, the chord of a -degree arc is
centimeters long, and the chord of a
-degree arc is
centimeters longer than the chord of a
-degree arc, where
The length of the chord of a
-degree arc is
centimeters, where
and
are positive integers. Find
Solution
![[asy] pointpen = black; pathpen = black+linewidth(0.7); pair A=(0,0), B=(0,22), C=OP(CR(A,11+165^.5),CR(B,22)), D=OP(CR(A,-9+165^.5),CR(C,22)); D(D(MP("A",A,E))--D(MP("B",B,N))--D(MP("C",C,W))--D(MP("D",D,SW))--A--C); D(circumcircle(A,B,C)); MP("22",(A+B)/2,E); MP("22",(C+B)/2,NW); MP("22",(C+D)/2,SW); MP("22",(A+B)/2,E); MP("x",(A+D)/2,SE); MP("x+20",(A+C)/2,NE); [/asy]](http://latex.artofproblemsolving.com/1/7/7/177b3a9528e0bd79c170d6bf594fde668c1e7b1a.png)
We let our chord of degree be
, of degree
be
, and of degree
be
. We are given that
. Let
. Since
, quadrilateral
is a cyclic isosceles trapezoid, and so
. By Ptolemy's Theorem, we have
Therefore, the answer is
.
See also
2001 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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