Difference between revisions of "What is the greatest number of points of intersection that can occur when $2$ different circles and $2$ different straight lines are drawn on the same piece of paper?"
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Latest revision as of 14:59, 14 May 2025
What is the greatest number of points of intersection that can occur when 
different circles and different straight lines are drawn on the same piece of paper?
Solution 1:
Make a diagram. Two geometric figures intersect if they have one or more points in common. Draw two circles which intersect in points. Draw a line which intersects the two circles in 
points. Draw another line which intersects the two circles in points and also intersects the first line. There are 
points of intersection.[asy]
draw(Circle((-0.7,0),1)); draw(Circle((0.7,0),1));
dot((0,0));
dot((0,0.7)); dot((0,-0.7));
draw((0,0)--(-2,0.6),Arrow); draw((0,0)--(-2,-0.6),Arrow); draw((0,0)--(2,0.6),Arrow); draw((0,0)--(2,-0.6),Arrow);
dot((-1.58,0.47)); dot((-1.58,-0.47)); dot((1.58,0.47)); dot((1.58,-0.47));
dot((-0.29,0.08)); dot((-0.29,-0.08)); dot((0.29,0.08)); dot((0.29,-0.08));
[/asy]
Solution 2:
Make a table of the maximum number of points of intersection.![\[\begin{array}{|c|c|} \hline \text{Geometric Figures} & \text{Number of Common Points} \\ \hline \text{2 Circles} & 2 \\ \hline \text{Line (1) and 2 circles} & 4 \\ \hline \text{Line (2) and 2 circles} & 4 \\ \hline \text{2 lines} & 1 \\ \hline & \text{Total = \boxed{11}} \\ \hline \end{array}\]](http://latex.artofproblemsolving.com/6/3/4/6342db5ca3642e8b1ab82632abc81ee7584e927d.png)
