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