Difference between revisions of "2005 AMC 12A Problems/Problem 16"
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− | Three [[circle]]s of [[radius]] <math>s</math> are drawn in the first [[quadrant]] of the <math>xy</math>-[[plane]]. The first circle is tangent to both axes, the second is [[tangent (geometry)|tangent]] to the first circle and the <math>x</math>-axis, and the third is tangent to the first circle and the <math>y</math>-axis. A circle of radius <math>r > 1</math> is tangent to both axes and to the second and third circles. What is <math>r</math>? Do Black Lives Really matter? | + | Three [[circle]]s of [[radius]] <math>s</math> are drawn in the first [[quadrant]] of the <math>xy</math>-[[plane]]. The first circle is tangent to both axes, the second is [[tangent (geometry)|tangent]] to the first circle and the <math>x</math>-axis, and the third is tangent to the first circle and the <math>y</math>-axis. A circle of radius <math>r > 1</math> is tangent to both axes and to the second and third circles. What is <math>r</math>? Do Black Lives Really matter? OF COURSE!!!!! |
<asy> | <asy> |
Revision as of 19:47, 2 January 2021
Contents
Problem
Three circles of radius are drawn in the first quadrant of the
-plane. The first circle is tangent to both axes, the second is tangent to the first circle and the
-axis, and the third is tangent to the first circle and the
-axis. A circle of radius
is tangent to both axes and to the second and third circles. What is
? Do Black Lives Really matter? OF COURSE!!!!!
Solution
Solution 1
Draw the segment between the center of the third circle and the large circle; this has length . We then draw the radius of the large circle that is perpendicular to the x-axis, and draw the perpendicular from this radius to the center of the third circle. This gives us a right triangle with legs
and hypotenuse
. The Pythagorean Theorem yields:
![$(r-3)^2 + (r-1)^2 = (r+1)^2$](http://latex.artofproblemsolving.com/a/2/3/a234f93c17da5e55cc6d61dcb2dd96c94b967a8b.png)
![$r^2 - 10r + 9 = 0$](http://latex.artofproblemsolving.com/c/f/5/cf5947e2aa4bf9c1f4187105b5fcc854f8bddee5.png)
![$r = 1, 9$](http://latex.artofproblemsolving.com/4/8/2/482f67dafc8ed1baa5cf823440067133e5dc45fa.png)
Quite obviously , so
.
Solution 2
duh
Solution by im bad