Difference between revisions of "2023 AMC 10A Problems/Problem 22"
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| − | 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{\textbf{(D) } \frac{3}{28}}</math> |
-andliu766 | -andliu766 | ||
Revision as of 20:46, 9 November 2023
Problem
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
Video Solution 1 by OmegaLearn
See Also
| 2023 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 21 |
Followed by Problem 23 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
