Difference between revisions of "2003 AMC 12A Problems/Problem 17"
(New page: == Problem == Square <math>ABCD</math> has sides of length <math>4</math>, and <math>M</math> is the midpoint of <math>\overline{CD}</math>. A circle with radius <math>2</math> and center ...) |
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<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac {16}{5} \qquad \textbf{(C)}\ \frac {13}{4} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {7}{2}</math> | <math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac {16}{5} \qquad \textbf{(C)}\ \frac {13}{4} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {7}{2}</math> | ||
+ | |||
+ | == Solution == | ||
+ | |||
+ | Let <math>D</math> be the origin. <math>A</math> is the point <math>(0,4)</math> and <math>M</math> is the point <math>(2,0)</math>. We are given the radius of the quarter circle and semicircle as <math>4</math> and <math>2</math>, respectively, so their equations, respectively, are: | ||
+ | |||
+ | <math>x^2 + (y-4)^2 = 4^2</math> | ||
+ | |||
+ | <math>(x-2)^2 + y^2 = 2^2</math> | ||
+ | |||
+ | |||
+ | Algebraically manipulating the second equation gives: | ||
+ | |||
+ | <math>y^2 = 2^2 - (x-2)^2</math> | ||
+ | |||
+ | <math>y^2 = (2-(x-2)(2+(x-2))</math> | ||
+ | |||
+ | <math>y^2 = (4-x)(x)</math> | ||
+ | |||
+ | <math>y = \sqrt{4x - x^2}</math> | ||
+ | |||
+ | |||
+ | Substituting this back into the first equation: | ||
+ | |||
+ | <math>x^2 + (\sqrt{4x - x^2} - 4)^2 = 4^2</math> | ||
+ | |||
+ | <math>x^2 + 4x - x^2 - 8\sqrt{4x - x^2} + 16 = 16</math> | ||
+ | |||
+ | <math>4x - 8\sqrt{4x - x^2} = 0</math> | ||
+ | |||
+ | <math>4x = 8\sqrt{4x - x^2}</math> | ||
+ | |||
+ | <math>16x^2 = 64(4x - x^2)</math> | ||
+ | |||
+ | <math>16x^2 = 256x - 64x^2</math> | ||
+ | |||
+ | <math>80x^2 - 256x = 0</math> | ||
+ | |||
+ | <math>x(80x - 256) = 0</math> | ||
+ | |||
+ | |||
+ | Solving each factor for 0 yields <math>x = 0 , \frac{16}{5}</math>. The first value of <math>0</math> is obviously referring to the point where the circles intersect at the origin, <math>D</math>, so the second value must be referring to the x value of the coordinate of <math>P</math>. Since <math>\overline{AD}</math> is the y-axis, the distance to it from <math>P</math> is the same as the x-value of the coordinate of <math>P</math>, so the distance from <math>P</math> to <math>\overline{AD}</math> is <math>\frac{16}{5} \Rightarrow B</math> |
Revision as of 19:47, 31 May 2008
Problem
Square has sides of length , and is the midpoint of . A circle with radius and center intersects a circle with raidus and center at points and . What is the distance from to ?
Solution
Let be the origin. is the point and is the point . We are given the radius of the quarter circle and semicircle as and , respectively, so their equations, respectively, are:
Algebraically manipulating the second equation gives:
Substituting this back into the first equation:
Solving each factor for 0 yields . The first value of is obviously referring to the point where the circles intersect at the origin, , so the second value must be referring to the x value of the coordinate of . Since is the y-axis, the distance to it from is the same as the x-value of the coordinate of , so the distance from to is