# Difference between revisions of "2003 AMC 12A Problems/Problem 17"

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== See Also == | == See Also == | ||

*[[2003 AMC 12A Problems]] | *[[2003 AMC 12A Problems]] | ||

− | *[[2003 AMC 12A/Problem 16|Previous Problem]] | + | *[[2003 AMC 12A Problems/Problem 16|Previous Problem]] |

− | *[[2003 AMC 12A/Problem 18|Next Problem]] | + | *[[2003 AMC 12A Problems/Problem 18|Next Problem]] |

## Revision as of 13:18, 1 January 2012

## Problem

Square has sides of length , and is the midpoint of . A circle with radius and center intersects a circle with radius 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 x-coordinate of the point where the circles intersect at the origin, , so the second value must be referring to the x 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