Difference between revisions of "2013 Mock AIME I Problems/Problem 1"
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== Solution == | == Solution == | ||
| − | 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{ | + | 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 == | ||
| + | * [[2013 Mock AIME I Problems]] | ||
| + | * Preceded by <math>\textbf{First Problem}</math> | ||
| + | * [[2013 Mock AIME I Problems/Problem 2|Followed by Problem 2]] | ||
Latest revision as of 09:01, 30 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
- 2013 Mock AIME I Problems
- Preceded by
- Followed by Problem 2
