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# Slicing a rectangular pyramid

Video transcript

So I have a three-dimensional
solid right over here. And I want to imagine
what type of a shape I would get if I were
to make a vertical cut. And just to refresh
ourselves, or give us a sense of what a
vertical cut is, imagine if this was made
out of jello or something kind of fairly soft. But it's still a
three-dimensional solid. And I were to make
a cut-- let's say I were to make a
cut right over here. So let's say I had
this big, sharp metal thing-- let me
draw it like that. So you have this big,
sharp metal thing. Let me draw it a
little bit neater. So you have this big,
sharp metal thing that I'm going to
cut right over here. So this is the thing that
I'm going to make the cut. And I'm going to
go straight down. This is a vertical cut
that we're talking about. So this is the thing that
I'm going to cut with. Let me make it big
enough so that it can capture the shape
that will result. So this thing right over
here, it's right in front. And I'm going to cut-- I'm
going to make it go straight down and cut through
this jello, or whatever you want to call it, this
rectangular pyramid of jello. And what would be the resulting
shape of the intersection between the jello and this
thing that I'm using to cut it? And now I encourage
you to pause your video and think about what the
resulting shape would be. And the shape would
be in two dimensions. Because this purple surface
is a two-dimensional-- you could view it
as part of a plane. And so where this
intersects when you cut down this rectangular pyramid is
the shape we're looking for. So I encourage you to pause
the video and think about it or try to come up
with it on your own. So let's think about it. And let me draw the
rectangular pyramid again. So that's the same one. And now let me see what
it would look like once I've done my cut, once I've
brought this thing down. So then this is where I cut. So I cut it right over here. And then it'll exit the bottom. It'll cut along
this side like that, cut along that side like that. And then it'll exit the
bottom right over there. And so let me draw
my whole thing. And so once I slice it down,
it will look like this. My best shot at drawing
it-- it will look like this. This is a vertical cut. So I've brought this thing down. And now the intersection between
the thing that I'm cutting with and this pyramid is going to
be this shape right over here. It cut into the top
right over there. It would get all the way to
the bottom right over there. And along this side, it
would cut right there. And along that side, it
would cut right over there. So what's the resulting
shape in two dimensions of essentially the
intersection between the slicer and the jello? Well, it would be this thing. It would look like a trapezoid. Let me do that in a new color. I'm overusing that one color. It would look like a trapezoid. So this would be
the resulting shape. So the resulting shape
would look like this. Just like that, if you made
the cut right over there.