My math homework is Impossible!?

Question

What is the greatest possible number of points of intersection of a square, a circle, and a triangle? Count all points of intersection with any two of the figures. (No side of the square is a side of the triangle) Can you please try to show your work?

WHEN 2 INTERSECT!!! NOT 3

Answer 1

Try it like this. The square can intersect the circle a maximum of eight times. This happens when the circle is wider than the square at the center but becomes narrower before it reaches the edge of the square, thus crossing it at two points on each edge. The triangle can intersect the circle a maximum of six times in the same way. The triangle can intersect the square six times as well, if you draw two legs of the triangle cutting off edges of the square and the third connecting them, going through opposite sides. It's hard to explain without drawing them. Anyway, the point is, look at the possible intersections of two shapes at a time, then add them together to get your answer, which in this case is 20.

Answer 2

First I will assume the triangle is a regular figure with equal length sides, an equilateral triangle. Consider the circle and square. Imagine the center of each coincide and the circle is larger and enclosed the square without touching. Imagine the circle shrinking. It first will contact the 4 corner points. As it gets smaller, the sides will each bisect a chord, which will make 8 intersections, 2 on each side, and then the circle will just fit inside the square with it touching the middle of the 4 sides and then if any smaller will be completely inside the square. Doing the same with the triangle gives me 3, 6 and 3. Lastly, the square and triangle. This one is more difficult, but as I visualize this, I started with the square larger than the triangle and shrinking, with no side parallel to a side of the triangle. As I see it, the end points of each end of the triangle can intersect only 6 points, 2 places on each side of the triangle. The triangle limits the maximum to 6. If I now overlay each, the most I can get is 8 + 6 + 6 = 20. There is no work to show. I have no way to draw you a picture. From what i have suggested you visualize, you should be able to make the drawings.

Answer 3

Mathematically I am not sure how to work this out, but logically it is pretty simple:

The maximum number of points the circle and square can intersect is 8. This is with a circle and square which have the same center, and the circle's diameter is slightly larger than the length of the squares sides. The circle will go outside the edges of the square, but the corners of the square will go outside the circle -- they intersect at two points on each side of the square.

The maximum points the triangle can intersect the square is 6. This is with the triangle having a bottom edge (horizontal) that is higher than the bottom edge of the square, and wider than the square, so that it pokes out either side of the square. The tip of the triangle is above the top edge of the square. This way the bottom side of the triangle cuts each of the sides of the square once (2 points), while each side of the triangle cuts a side and the top of the square (4 points) for a total of 6 points of intersection.

The maximum points the the triangle can intersect the circle is also 6. This is just like the first case, witht he triangle and circle having the same center and the points of the triangle stick out further than the edge of the circle -- the circle will cut each side of the triangle twice, for a total of 6 points of intersection.

If you had to figure out how many points all 3 had to intersect, I think the answer is four: The top edge of the square will intersect the 2 points where the square and triangle intersect, and then the sides of the square will come down to intersect the diagonal sides of the triangle where the circle intersects it. The bottom edge of the square would have to be the same as the bottom side of the triangle to intersect all four of the lower intersections so we can't do that, meaning the square would have to have its bottom edge either above or below the bottom edge of the triangle.

Answer 4

Square and circle: 8
Circle and triangle: 6
Triangle and Square: 6

Answer 5

Square and circle: 4
Circle and triangle: 3
Triangle and Square:3

Answer 6

infinite solutions