Difference between revisions of "2023 AMC 10A Problems/Problem 22"
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<math>\textbf{(A) } \frac{1}{14} \qquad \textbf{(B) } \frac{1}{12} \qquad \textbf{(C) } \frac{1}{10} \qquad \textbf{(D) } \frac{3}{28} \qquad \textbf{(E) } \frac{1}{9}</math> | <math>\textbf{(A) } \frac{1}{14} \qquad \textbf{(B) } \frac{1}{12} \qquad \textbf{(C) } \frac{1}{10} \qquad \textbf{(D) } \frac{3}{28} \qquad \textbf{(E) } \frac{1}{9}</math> | ||
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| + | ==Solution== | ||
| + | Connect the centers of <math>C_1</math> and <math>C_4</math>, and the centers of <math>C_3</math> and <math>C_4</math>. Let the radius of <math>C_4</math> be <math>r</math>. Then, from the auxillary lines, we get <math>(\frac{1}{4})^2 + (\frac{3}{4}+r)^2 = (1-r)^2</math>. Solving, we get <math>r = \boxed{\frac{3}{28}}</math> | ||
== Video Solution 1 by OmegaLearn == | == Video Solution 1 by OmegaLearn == | ||
https://youtu.be/jcHeJXs9Sdw | https://youtu.be/jcHeJXs9Sdw | ||
Revision as of 16:12, 9 November 2023
Circle 
and 
each have radius 
, and the distance between their centers is 
. Circle 
is the largest circle internally tangent to both and . Circle 
is internally tangent to both and and externally tangent to . What is the radius of ?
Solution
Connect the centers of and , and the centers of and . Let the radius of be 
. Then, from the auxillary lines, we get 
. Solving, we get 
