Difference between revisions of "2001 AMC 10 Problems/Problem 4"
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<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> | ||
| − | + | ==Solution== | |
| − | == Solution 1 == | + | === Solution 1 === |
Here is the picture: http://www.artofproblemsolving.com/Forum/download/file.php?id=6658& | Here is the picture: http://www.artofproblemsolving.com/Forum/download/file.php?id=6658& | ||
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Therefore, <math> 2 \times 3 = \boxed{\textbf{(E) }6} </math>. | Therefore, <math> 2 \times 3 = \boxed{\textbf{(E) }6} </math>. | ||
| − | == Solution 2 == | + | === Solution 2 === |
We know that the maximum amount of points that a circle and a line segment can intersect is <math>2</math>. Therefore, because there are <math>3</math> line segments in a triangle, the maximum amount of points of intersection is <math>\boxed{\textbf{(E) }6}</math>. | We know that the maximum amount of points that a circle and a line segment can intersect is <math>2</math>. Therefore, because there are <math>3</math> line segments in a triangle, the maximum amount of points of intersection is <math>\boxed{\textbf{(E) }6}</math>. | ||
Revision as of 18:33, 13 June 2011
Problem
What is the maximum number of possible points of intersection of a circle and a triangle?
Solution
Solution 1
Here is the picture: http://www.artofproblemsolving.com/Forum/download/file.php?id=6658&
We can draw a circle and a triangle, such that each side is tangent to the circle. This means that each side would intersect the circle at one point.
You would then have 
points, but what if the circle was bigger? Then, each side would intersect the circle at 2 points.
Therefore, 
.
Solution 2
We know that the maximum amount of points that a circle and a line segment can intersect is 
. Therefore, because there are line segments in a triangle, the maximum amount of points of intersection is 
.
See Also
| 2001 AMC 10 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
| 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 10 Problems and Solutions | ||
