Difference between revisions of "2013 Mock AIME I Problems/Problem 1"
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Let <math>A_3</math> be the center of circle <math>C_3</math> and <math>Q</math> be the point of tangency between <math>C_3</math> and <math>C_2</math>. Note that triangles <math>PQA_2</math> and <math>A_3PA_2</math> are similar, so <math>\frac{A_3A_2}{PA_2}=\frac{PA_2}{A_2Q}=3</math> and <math>A_3A_2=r+1=9</math>. Thus the radius of <math>C_3</math> is <math>\boxed{008}</math>. | Let <math>A_3</math> be the center of circle <math>C_3</math> and <math>Q</math> be the point of tangency between <math>C_3</math> and <math>C_2</math>. Note that triangles <math>PQA_2</math> and <math>A_3PA_2</math> are similar, so <math>\frac{A_3A_2}{PA_2}=\frac{PA_2}{A_2Q}=3</math> and <math>A_3A_2=r+1=9</math>. Thus the radius of <math>C_3</math> is <math>\boxed{008}</math>. | ||
+ | |||
+ | == See Also == | ||
+ | * Preceded by <math>\textbf{First Problem}</math> | ||
+ | * [[2013 Mock AIME I Problems/Problem 2|Followed by Problem 2]] |
Revision as of 18:34, 29 July 2024
Problem 1
Two circles and , each of unit radius, have centers and such that . Let be the midpoint of and let be a circle externally tangent to both and . and have a common tangent that passes through . If this tangent is also a common tangent to and , find the radius of circle .
Solution
Let be the center of circle and be the point of tangency between and . Note that triangles and are similar, so and . Thus the radius of is .
See Also
- Preceded by
- Followed by Problem 2