Slicing a Right Rectangular Pyramid with a Plane

Ah, there's nothing like the peaceful beauty of camping in the great outdoors. Wait a minute. It's the brothers Tom and Eddy and things are not peaceful! If they are going to share this tent they each need their own personal space. To divide their tent into two rooms, they will need to practice slicing a right rectangular pyramid with a plane. Here's their tent, which is in the shape of a pyramid. The floor, which is a rectangle, is the base of our pyramid. There are four lateral faces each of which is an isoceles triangle. The four triangles meet at a point called the apex. The apex is directly above the center of the base, so we call this a right rectangular pyramid. How can we divide this pyramid so that each brother has room to relax? We need to slice it. There are lots of ways to slice the pyramid, each of which create a plane section. What shape is created if we slice parallel to the base? That means there is a plane intersecting the pyramid. The resulting plane section is a rectangle. Depending on where we slice the pyramid, we can create smaller or larger rectangles but these rectangles will always be proportional to the base. That means the ratio of the length to the width of the plane section, is equal to the ratio of the length to the width of the base. That's pretty cool, but would this be a practical way to divide the tent? Only if you want it to be a double-decker. Another option for slicing the pyramid would be perpendicular to the base. What shape do you think will be created here? Let's rotate the pyramid to get a better look. There is exactly one pair of parallel sides so this is a trapezoid. By moving the plane we can make the trapezoid bigger or smaller. Do you think all these trapezoids are proportional to each other? Because the base on the bottom always stays the same length while the top changes they are not proportional to each other. Do you think these trapezoids are a good way to divide the tent? Only if one of the brothers is ok having the smaller room, since this slicing doesn't divide the pyramid equally. How can we divide the pyramid into equal parts? A slice through the apex perpendicular to the base! Note that when we look at the base, the slice is also parallel to two edges of it. What is the resulting plane-section? A triangle! In fact, it's an isoceles triangle, since these two sides are equal. Additionally, the height of the triangle is the height of the pyramid. An isoceles traingular curtain is a perfect solution for the brothers' tent. While Tom and Eddy are dividing their tent, let's review what we've learned. There are many ways to slice a right rectangular pyramid. We can slice parallel to the base resulting in a rectangular plane-section. We can slice perpendicular to the base and parallel to an edge of the base giving us a trapezoid unless that slice goes through the apex, which gives us an isosceles triangle. Tom and Eddy finally have their own spaces in the tent. But, wait a minute, what's that sound? With all the wildlife around, the brothers really should stick together!