A Math Forum Project

Geometry Forum - Problem of the Week Solutions - Boutros' Geometry Questions, Oct. 17-21, 1994 Annie says: We were very excited about the responses to this relatively random set of questions. There was a great deal of imagination exercised, both in visualizing the geometry (we especially liked Micah Derr's donut image) and in inventing questions for the rest of us to answer. We encourage readers to browse these challenge questions (answers to #5). We plan to use some of them in future POWs. The students from Mount St. Joseph's and Scott Hess realized that a triangle can also share three points with a plane if it has just one side in the plane. Scott Hess Number one...Yes a plane can intersect a circle in two points, by simply going through the middle. Yes a plane can intersect a circle in one point. It would look similar to a hoola-hoop standing up on a street. Finally, if a circle and a plane share three points they must be coplaner (the circle must be on the plane). Number two... Yes a triangle can share two points with a plane, the same way a circle can. Yes a triangle can share one point with a plane, but that point would have to be one of the vertices. The answer to the last part is a little more complicated than with the circle. If three points of the triangle intersect with the plane, then either one side of the triangle is on the plane, or the whole triangle is on the plane, but two sides of the triangle cannot be on the same plane while the other is not. Number three... you could place the segments from end point to end point, but you would need to use an infinite number of them. It would be like trying to construct space from points. Oh, I admit it could be done, but it would take an infinite number of points, and just wouldn't be practical. Number four... If n = 4 than T(# of triangles) is 2, if n = 5 then T = 3, if n = 6 then T = 4, if n = 7 then T = 5. Using these examples, I came up with a theory that n - 2 = T, where n is # of sides and T is lowest possible # of triangles. Number five... Can three points of a polygon intersect a plane if none of the sides are on that plane? How about a polyhedron? Finally, can a polyhedron share exactly two points with a plane? Micah Derr 1. A plane can intersect a circle in exactly 2 points. This happens if the circle breaks the plane of the plane. Ex. Like when you are dunking a doughnut in milk. The doughnut is halfway in the milk and only at 2 points does the doughnut intersect the milk. A circle can also intersect a plane in one point if and only if the circle is standing up on the plane. If the circle and the plane have three points in common then the circle lies in the plane. 2. A triangle can interesect a plane in exactly 2 points. This can happen when the plane cuts off the top of the triangle. A triangle can interect a plane in one point if the triangle is standing, on one point, on the plane. If the triangle and the plane have three distinct points in common then the triangle lies on the plane. 3. The only way I can come up with to use segments to make a ray is simply this: to put an infinite amount of segments together in one direction because a ray extends forever in one direction. 4. If you don't know what n is you can use this formula: x = number of sides, n = x - 2 Example: sides triangles 4 2 5 3 6 4 7 5 5. What is the fewest amount of planes needed to divide space into nine regions? What is greatest amount of planes needed to divide space into nine regions? Erik Ostrowski 1. A plane can intersect a circle in exactly two points or in exactly one point (if the edge of the circle just touches the plane). If the plane and the circle have three points in common, then the circle lies on the plane. 2. A plane can intersect a triangle in exactly two points or exactly one point (if a corner of the triangle just touches the plane). If the plane and the triangle have three points in common, then the triangle lies on the plane. 3. Segments cannot be put together to form a ray because they will always have two endpoints. A true ray only has one endpoint. But if you have an infinite amount of segments going in one direction, then a ray could be formed. 4. The solution for #4 can be explained by a chart and an equation: # of sides | # of triangles 4 | 2 5 | 3 6 | 4 7 | 5 8 | 6 the # of triangles = n-2 where n = number of sides on n-gon 5. Name the least number of sections a plane can be divided into by n number of lines. Jennifer Abrahamsen 1. If one part of a circle is on one side of a plane and the other part on the other side of the plane, then a circle can intersect a plane in exactly 2 points. If the circle is resting on its side (point) on the plane, then it can intersect a plane at exactly 1 point. If the plane and the circle have 3 points in common, then the circle is on the plane. 2. If part of the triangle is on one side of the plane and the other part is on the other side, then yes a triangle can intersect a plane in exactly two points. If a vertex of the triangle is resting on the plane, then it can intersect a plane at exactly 1 point. If the plane and the triangle have 3 points in common, then the triangle is on the plane. 3. You would need an infinite amount of segments. This is because one side of a ray, according to its definition, must continue into infinity. 4. # of sides Least # of triangles 4 2 5 3 6 4 7 5 n n - 2 5. How many different planes can be drawn for 10 noncollinear points if they are taken 3 at a time. No more than two are collinear. Try and find a formula for this. Eric Wahl 1. yes; yes; circle must lie in the plane 2. yes; yes; triangle must lie in the plane 3. You can join segments together in a line (end to end) and use one end as the endpoint of the ray, but you'd need an infinite number of segments to do this. 4. Assuming zero is not an appropriate response, if problem assumes the region is divided into triangular regions, the minimum would be n-2. n minimum triangles 4 2 5 3 6 4 7 5 n n - 2 5. If you take the same n-gon as in number 4, what is the smallest number of isosceles triangles you can divide it into if ther n-gon is regular? (ans. n isosceles triangles) Mount St. Joseph Academy 1. A plane can intersect a circle in exactly 2 points (the circle is passing through the plane); in exactly 1 point (the circle and the plane are tangent); and to have 3 points in common, the circle is on the plane. 2. A plane can intersect a triangle in exactly 2 points (the plane is passing through the triangle); in exactly 1 point (the plane is touching the tip of the triangle); and if they have 3 distinct points in common, the plane is either intersecting a whole side of the triangle or the triangle is lying on the plane. 3. You would need an infinite number of segments to make a ray because a ray goes on forever in one direction. 4. If you draw all the diagonals from one vertex, then if n = 4 you get 2 triangles, if n = 5 -- 3 triangles, n = 6 -- 4 triangles, n = 7 -- 5 triangles. A formula would be if n = the number of sides of the polygon, then n - 2 triangles are formed when drawing all the diagonals from one vertex. 5. Erin Foley's Question: If you know the measure of an exterior angle of a triangle, can you find out the 2 remote interior angles? If you know the 2 remote interior angles, can you find out the remote exterior angle? Erin Foley's 2nd Question: Why do the exterior angles of a polygon always sum to 360 regardless of the number of sides of the polygon? Give me a convincing argument. Susanna Puntel's Question: What is the measure of one interior angle of a 4-sided regular polygon? 5-sided? 6-sided? n-sided? Write a formula showing how to find the measure of one interior angle of an n-sided regular polygon. Kim Biedermann's Question: Could a line and a circle intersect in exactly 2 points? 1 point? 3 points? Maria Szczesniak's Question: Can the sum of the interior angles of a polygon equal 90? Why or why not? Sarah Egner's Question: Can a plane intersect another plane at one point? Moira Conway's Question: Name some of the ways to prove two angles are congruent, for example, they are alternate interior angles formed by parallel lines. Monika Krzyspiak's Question: What is the sum of the exterior angles of a triangle? quadrilateral? pentagon? hexagon? decagon? What conclusion can you draw from your answers? Trish Ritter's Question: Can a plane intersect a rhombus in exactly 2 points? Can a plane bisect any angle of a rhombus? Fairfield HS 5. 2 noncollinear rays with a common endpoint form 1 angle. 3 rays form 3 angles 4 rays form 6 angles 5 rays form __?__ angles 6 rays form __?__ angles 12 rays form __?__ angles Can you find a pattern? If so, what is the pattern. Jessica Muniz 5. Can two angles intersect in three points? David Gluckman 5. Can a trapezoid intersect a plane in exactly 2 points? Can a trapezoid intersect a plane in exactly 1 point? Can a trapezoid intersect a plane in exactly 3 points? Kim Carbone 5. How many possible diagonals does a 4-gon have, a 5-gon, 6-gon? Is there a pattern to find out how many diagonals an n-gon has? Samantha Brenner 5. If four planes intersect each other, can you draw one line that is parallel to all of them? Cecilia Merediz 5. If you had an n-gon with n = 3 and you cut across and cut off 2 of the corners, how many corners will you have left? What if n = 4? n = 5? Come up with a formula if you don't know 'n'. Brad Warner 5. Can a plane intersect a line in exactly 2 points? In exactly 1 point? What happens if the plane and the line have 3 points in common? Previous page || Next problem || Previous problem || Table of Contents || Forum Home Page