Difference between revisions of "2015 AMC 12A Problems/Problem 11"
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==Problem== | ==Problem== | ||
| − | On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k | + | On a sheet of paper, Isabella draws a circle of radius <math>2</math>, a circle of radius <math>3</math>, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly <math>k \ge 0</math> lines. How many different values of <math>k</math> are possible? |
<math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}}\ 5\qquad\textbf{(E)}\ 6</math> | <math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}}\ 5\qquad\textbf{(E)}\ 6</math> | ||
Revision as of 00:35, 6 February 2015
Problem
On a sheet of paper, Isabella draws a circle of radius 
, a circle of radius 
, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly 
lines. How many different values of 
are possible?
$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}}\ 5\qquad\textbf{(E)}\ 6$ (Error compiling LaTeX. Unknown error_msg)
Solution
Isabella can get 
lines if the circles are concentric, 
if internally tangent, if overlapping, if externally tangent, and 
if non-overlapping and not externally tangent. There are 
values of .
See Also
| 2015 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 10 |
Followed by Problem 12 |
| 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 12 Problems and Solutions | |
