Difference between revisions of "2023 AMC 10A Problems/Problem 22"
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==Solution== | ==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> | 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> | ||
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+ | -andliu766 | ||
== Video Solution 1 by OmegaLearn == | == Video Solution 1 by OmegaLearn == | ||
https://youtu.be/jcHeJXs9Sdw | https://youtu.be/jcHeJXs9Sdw |
Revision as of 16:24, 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
-andliu766