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) | + | <math> \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math> |
==Solution== | ==Solution== | ||
Latest revision as of 12:24, 29 March 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?
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 | |
