Difference between revisions of "2001 AIME I Problems/Problem 13"
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== Problem == | == Problem == | ||
− | In a certain circle, the chord of a <math>d</math>-degree arc is 22 centimeters long, and the chord of a <math>2d</math>-degree arc is 20 centimeters longer than the chord of a <math>3d</math>-degree arc, where <math>d < 120.</math> The length of the chord of a <math>3d</math>-degree arc is <math>- m + \sqrt {n}</math> centimeters, where <math>m</math> and <math>n</math> are positive integers. Find <math>m + n.</math> | + | In a certain [[circle]], the [[chord]] of a <math>d</math>-degree arc is <math>22</math> centimeters long, and the chord of a <math>2d</math>-degree arc is <math>20</math> centimeters longer than the chord of a <math>3d</math>-degree arc, where <math>d < 120.</math> The length of the chord of a <math>3d</math>-degree arc is <math>- m + \sqrt {n}</math> centimeters, where <math>m</math> and <math>n</math> are positive integers. Find <math>m + n.</math> |
== Solution == | == Solution == | ||
− | {{ | + | {{image}} |
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+ | We let our chord of degree <math>d</math> be <math>\overline{AB}</math>, of degree <math>2d</math> be <math>\overline{AC}</math>, and of degree <math>3d</math> be <math>\overline{AD}</math>. We are given that <math>AC = AD + 20</math>. Since <math>AB = BC = CD = 22</math>, quadrilateral <math>ABCD</math> is a [[cyclic quadrilateral|cyclic]] [[isosceles trapezoid]], and so <math>BD = AC = AD + 20</math>. By [[Ptolemy's Theorem]], we have | ||
+ | <cmath>\begin{align*} | ||
+ | AB \cdot CD + AD \cdot BD &= AC \cdot BD\ | ||
+ | 22^2 + 22x = (x+20)^2 &\Longrightarrow x^2 + 18x - 84 = 0\ | ||
+ | x = \frac{-18 + \sqrt{18^2 + 4\cdot 84}}{2} &= -9 + \sqrt{165}\end{align*}</cmath> | ||
+ | Therefore, the answer is <math>m+n = \boxed{174}</math>. | ||
== See also == | == See also == | ||
{{AIME box|year=2001|n=I|num-b=12|num-a=14}} | {{AIME box|year=2001|n=I|num-b=12|num-a=14}} | ||
+ | |||
+ | [[Category:Intermediate Geometry Problems]] |
Revision as of 20:19, 13 June 2008
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
An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.
We let our chord of degree be , of degree be , and of degree be . We are given that . 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 |