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 12:18, 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