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Cross Sections of Prisms
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00:03: Let's review some cross-sections that we come across pretty often in Math. Remember that a cross-section, a two-dimensional cross section is the intersection of a plane and a three-dimensional figure. So we have a set of figures here and in each case a plane being drawn that is essentially cutting through the three-dimensional figure, and the idea is, what is the shape of the part of the plane that is shared by the plane and the figure? So to take the first example here, we have a triangular prism. And you noticed that it's being cut by a plane and that is parallel to the base. Sorry, actually it's cutting through the base because the base of this Prism is triangular. So this is actually one of the bases and the other base is on the other side, we can't see it. So this plane is cutting through its perpendicular in a sense to the bases. If you imagine what you would see if you were to take the top of that prism off, the part that's above the plane, you would essentially just see a rectangle. Now, on the other hand, if we were to cut this triangular prism, with a plane that ran parallel to the basis, so it was a plane like that cut through this way and that we opened it up, then we would actually have a triangle. 01:44: This next picture is of a pyramid, it's a square pyramid, well, presumably a square, but it's definitely got a rectangular base. And if you imagine what you would see if you took the part of the pyramid that is in front of the plane here, that's cutting it, and looked at the shape, you would see a triangle. Now, of course, if we were to cut this pyramid with a plane that was parallel to the base, then what we would get is, well, if the base is square, we would get a square and if the base is some other rectangle then we would just get a rectangle that would be that. Well, it wouldn't necessarily be the size of the base, but it'd be similar to the base. This next one is a cube and so we're just gonna declare that it's a cube. So the cross-section that you get, if you cut it parallel to any of the faces is you're gonna get just a square. And of course, in the case of a cube, there's really not a distinction any face connect as the base, so it really doesn't matter which face we cut the cube parallel with, we'll get the square. 03:01: In this case here, we have a three-dimensional essentially rectangular prism and if we cut it parallel to the base, that's shown, we would get kind of a long thin rectangle. Obviously, all of these would be, they're fairly thin if we cut it, say parallel to this base here like this, we would get another one. If we were to cut it parallel to this face here, then we would get kind of a fatter rectangle. Again, in all these cases, we're cutting very carefully, we're not cutting obliquely, that is, we're not cutting it strange angles, we're cutting either parallel to a base or parallel to some specified face. The last case here is supposed to represent a sphere. Sphere is a special case. It really doesn't matter how you, well, slice it. You will get a circular cross-section no matter what. You can see, even if you cut off just a piece of the sphere, if you were to look at that orange, that orange shape that you get, you would see it was a circle, it would just be a smaller circle. And of course the largest circle you could ever get would be one with the same radius as the sphere itself. And that would be if you cut it straight through the center of the sphere. Otherwise you're just gonna get smaller and smaller circles as you move away from the center.
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Question 1
Identify the cross section from slicing the figure vertically.
Responses
A triangletriangle
B rectanglerectangle
C rectangular prismrectangular prism
Question 2
Identify the cross section from slicing the figure vertically.
Responses
A triangletriangle
B rectanglerectangle
C triangular prismtriangular prism
Question 3
Identify the cross section from slicing the figure horizontally.
Responses
A triangletriangle
B rectanglerectangle
C rectangular prismrectangular prism
1 answer
Based on the description provided in the transcript, here's how to approach the questions regarding the cross sections:
Question 1:
Identify the cross section from slicing the figure vertically.
Responses:
A: triangle
B: rectangle
C: rectangular prism
From the description, if the figure being referenced is a triangular prism and we slice it vertically, the cross section will reveal a rectangle (B).
Question 2:
Identify the cross section from slicing the figure vertically.
Responses:
A: triangle
B: rectangle
C: triangular prism
If the figure is a pyramid with a rectangular base and we slice it vertically, the resulting cross section will be a triangle (A).
Question 3:
Identify the cross section from slicing the figure horizontally.
Responses:
A: triangle
B: rectangle
C: rectangular prism
If the figure is a triangular prism and we slice it horizontally (parallel to the bases), the cross section will be a triangle (A).
