Difference between revisions of "2011 AIME II Problems/Problem 10"
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== Problem 10 == | == Problem 10 == | ||
− | A circle with center O has radius 25. Chord <math>\overline{AB}</math> of length 30 and chord <math>\overline{CD}</math> of length 14 intersect at point P. The distance between the | + | A [[circle]] with center <math>O</math> has radius 25. [[Chord]] <math>\overline{AB}</math> of length 30 and chord <math>\overline{CD}</math> of length 14 intersect at point <math>P</math>. The distance between the [[midpoint]]s of the two chords is 12. The quantity <math>OP^2</math> can be represented as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are [[relatively prime]] positive integers. Find the remainder when <math>m + n</math> is divided by 1000. |
== Solution == | == Solution == | ||
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Let <math>x</math>, <math>a</math>, and <math>b</math> be lengths <math>OP</math>, <math>EP</math>, and <math>FP</math>, respectively. OEP and OFP are also right triangles, so <math>x^2 = a^2 + 20^2 \to a^2 = x^2 - 400</math>, and <math>x^2 = b^2 + 24^2 \to b^2 = x^2 - 576</math> | Let <math>x</math>, <math>a</math>, and <math>b</math> be lengths <math>OP</math>, <math>EP</math>, and <math>FP</math>, respectively. OEP and OFP are also right triangles, so <math>x^2 = a^2 + 20^2 \to a^2 = x^2 - 400</math>, and <math>x^2 = b^2 + 24^2 \to b^2 = x^2 - 576</math> | ||
− | We are given that <math>EF</math> has length 12, so, using the Law of Cosines with <math>\triangle EPF</math>: | + | We are given that <math>EF</math> has length 12, so, using the [[Law of Cosines]] with <math>\triangle EPF</math>: |
<math>12^2 = a^2 + b^2 - 2ab \cos (\angle EPF) = a^2 + b^2 - 2ab \cos (\angle EPO + \angle FPO)</math> | <math>12^2 = a^2 + b^2 - 2ab \cos (\angle EPF) = a^2 + b^2 - 2ab \cos (\angle EPO + \angle FPO)</math> | ||
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Square both sides: <math>169 x^4 - 187000 x^2 + 51,840,000 = 225 x^4 - 219600 x^2 + 51840000</math> | Square both sides: <math>169 x^4 - 187000 x^2 + 51,840,000 = 225 x^4 - 219600 x^2 + 51840000</math> | ||
− | This reduces to <math>x^2 = \frac{4050}{7} = (OP)^2</math>; (4050 + 7) divided by 1000 has remainder <math>\ | + | This reduces to <math>x^2 = \frac{4050}{7} = (OP)^2</math>; (4050 + 7) divided by 1000 has remainder <math>\boxed{057}</math>. |
+ | |||
+ | ==See also== | ||
+ | {{AIME box|year=2011|n=II|num-b=9|num-a=11}} | ||
+ | |||
+ | [[Category:Intermediate Geometry Problems]] |
Revision as of 22:32, 22 August 2011
Problem 10
A circle with center has radius 25. Chord of length 30 and chord of length 14 intersect at point . The distance between the midpoints of the two chords is 12. The quantity can be represented as , where and are relatively prime positive integers. Find the remainder when is divided by 1000.
Solution
Let and be the midpoints of and , respectively, such that intersects .
Since and are midpoints, and .
and are located on the circumference of the circle, so .
The line through the midpoint of a chord of a circle and the center of that circle is perpendicular to that chord, so and are right triangles (with and being the right angles). By the Pythagorean Theorem, , and .
Let , , and be lengths , , and , respectively. OEP and OFP are also right triangles, so , and
We are given that has length 12, so, using the Law of Cosines with :
Substituting for and , and applying the Cosine of Sum formula:
and are acute angles in right triangles, so substitute opposite/hypotenuse for sines and adjacent/hypotenuse for cosines:
Combine terms and multiply both sides by : $144 x^2 = 2 x^4 - 976 x^2 - 2 (x^2 - 400) (x^2 - 576) + 960 \sqrt{(x^2 - 400)(x^2 - 576)$ (Error compiling LaTeX. ! Missing } inserted.)
Combine terms again, and divide both sides by 64:
Square both sides:
This reduces to ; (4050 + 7) divided by 1000 has remainder .
See also
2011 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 |