Hi everyone, today I'm here with who might be a familiar face. It's Diana from physics girl
- Hello! - Thanks for being on my channel
We thought that today we'd do a bit of an exam
unboxing which is, you know, a common thing around here,
but we're going to look at another very old exam. So, we have here the MIT entrance exam from
1876 and we're just going to look at the geometry section of it.
Now, Diana makes for an interesting addition to this video because she actually went to MIT,
is an MIT graduate, so it'll be interesting to see if you know she can pass the entrance test.
I got in much later than 1876.
And in fact much later than 1976.
So I'm sure the requirements have changed.
I also love that the exam says arithmetic, algebra and geometry like math of the only thing you needed to know.
That's all there is, there's no calculus involved.
No, of course not. There's also no engineering. No physics. No chemistry. No bio knowledge. Just math.
Yeah, so I guess an interesting question would be you know
What did you have to personally do to get into MIT? You probably didn't have to do this?
No, I didn't have to do
This is so funny, I can't wait until
We will show you in a second.
I can't wait till you guys see this example cuz hilarious.
We had SATs, so you had to take those which had writing, math and reading comprehension.
Then you also to take subject tests.
So I had to take the physics subject test,
which was not easy and I had to study pretty hard and specifically for
because our physics classes at my tiny private school that I went to high school that only went up to
mechanics didn't go passed into any E&M but there was E&M on this subject test.
There was nothing calculus based.
Phew.
Yeah, just the standardized tests.
- All right, so you really had to prove yourself in physics to actually get in. - Yes, I did.
I think when we show you this exam, in 1876 that was only a few years after
MIT came into existence and I think they just wanted to see that people had a basic education and
some mathematics and I think it was more, you know,
there were only a few people that could maybe afford to go to MIT or or just
had the opportunity to go. So probably much less competitive to get in,
so maybe the exam reflects that. But there's some interesting vocab we'll show you through.
Yeah, I mean this honestly
all your comments are making me wonder what it was like at the time? Was it really
I assume a lot of people didn't know about MIT in the way that they
internationally know about it now.
Or there just weren't as many people applying. Now people apply from all over the country and all over the world
So it's a bit more rigorous.
Yeah.
Alright, let's have a look.
Ooh! this is gonna be fun.
All right, so this is what the exam looks like. We've got seven questions on here, this is just geometry.
But having a little read through them, some of them it's not even quite clear what they're asking.
So I'll read you the first question:
"If a straight line drawn parallel to the base of a triangle
bisects one of the sides, it bisects the other also; and the portion of it
intercepted between the two sides is equal to one-half the base."
yeah, so
So where's the question?
It's like okay.
There's not even a question mark or even like a statement of: okay do this now.
Yeah, so I guess the assumption here is that you have to prove the latter half of this.
But it's interesting that they don't write proof,
because they do actually, in these later questions,
write "prove". Like, "prove the side of the regular inscribed hexagon is equal to the radius of the circle."
Okay, this one is just-- first of all
it's like they tell you, "if this then that. And then also this happens. And then also this happens."
so it feels very much like
Like here's some statements
It really doesn't feel like this is something you should prove
but I can't imagine anything else you would do with it besides prove?
I'm surprised at how short this is. Well, I guess okay
so there's the arithmetic, algebra and then we're just on geometry. Skip number six.
So yeah, number six it looks like,
something about a polar triangle, whatever that is, and then, see figure on blackboard.
So we're missing a bit of information there
but the rest of these I guess we have enough knowledge to prove them and it looks like
To me, it seems like it's using school concepts that I remember doing at school. So having a look at these
like I don't
particularly ever remember doing
questions like this, like defining a frustum, whatever that is?
Yeah
But I do remember, I think, the laws behind some of these geometry questions and they're really basic things like that
angles in a triangle add up to 180 and that angles on a line add up to 180 and
That if you have parallel lines and draw a line between them
The corresponding angles you get are equal to each other
Mm-hmm. Yeah, I mean, it's just looking at number two, "the diameter
perpendicular to a chord"
so first of all
you have to know some of these terms like you've to remember what a chord is and I
think I remember that that just means if you have a circle and you have a line going
from one part of the circle through the circle to the other part, that's a chord.
Yeah. I think it's it's tackling
vocab here. So we've got chord, we've got bisects
Mm-hmm. I saw that pop up. Yeah bisects, so that's splitting a line, I think, equally
in half, I believe.
Secant is not actually a word I particularly remember but I know I've seen before.
I did not remember what that was
I thought it was like a mariner term
yeah, maybe that's
I don't know.
Although, considering it's from 1876, I think
one thing that is notable is that math doesn't really change that much.
They are still things that seem doable to me, you know, with a modern math education.
It's not like,
fundamentally, these concepts are changing and I think because
the questions are reasonably simple, we still can just figure them out
That's pretty interesting. I hadn't thought about that. About the fact that this is from 1876 and even still
reading this sentence I understand basically what they're saying.
I don't use the word subtended on a daily basis, but I can get you know
I can get the point of what they're trying to say. And again, they're just stating something here
But it looks like something they want you to prove.
Yeah, if anything it's surprisingly understandable. I actually think that if we were doing an 1876 English exam,
I would struggle more.
Oh for sure. I think if we were doing a
2019 English exam I'd struggle more.
So let's just solve a couple of the questions to have a look at what the answers basically look like for these things.
Yeah, I'm ready. Let's do it.
Okay
So I'm going to take charge of solving the first problem on here and I read it out
before but we will draw a little diagram
so we have
a triangle
Forgive me for my drawing skills, I'm leaning over the tripod
We have a straight line drawn parallel to the base and it bisects one other side
So, okay. We want to come in halfway between these
I need a ruler.
But that's fine.
Looks good to me.
"Bisects the other also", sure, "and the portion of it intercepted between the two sides" is
That part was the most confusing to me but they just mean
- I guess they mean this part in the middle. - Yeah.
"Intercepted between the two sides",
is not the phrase I would choose to use, but sure.
the portion between, on the line between the two sides.
Okay that's equal to half of this line. So yeah, let's draw in a few more lines to help us out
So I'm going to draw another parallel line
through the tip there.
We can put one there as well.
The cool thing about parallel lines that I remember from school geometry is that
if they're parallel and we have a line between them,
then
Well, I think it's the alternating
angles are the same.
Yeah that, these ones but then also these ones.
Yeah.
There's gotta be a name for those angles.
So it's this angle here and this angle here, like the interior angles
are the same so those are the same and then,
by a similar reasoning this angle and this angle here will also be the same as each other.
- Yep. - So we can go from there
Because this line is equal to this line, and also
this line is equal to that line, what we're getting at here is we've got some similar triangles going on. In this case we have
this triangle here would be similar to the larger triangle
So by saying similar, I mean that it's kind of just the same but scaled down a bit
So the ratios between the sides and the angles should be the same for both those triangles.
So that's what I'm going to work with to say that if drawing this line across here has cut this line in half
then
Because they're similar and they related by a ratio of one half then I'm going to think that applies to this
Line on the bottom as well, that, because they're similar triangles. These lines would be related by 1/2
So I know I haven't actually written anything down
But I did look up similar triangles.
Because I didn't remember what the terms of similar triangle is but it gives you that the triangles have congruent angles
Which means that they have essentially the same angles
But you were talking about how you've got these same angles here and these same angles here. And then this is obviously sharing things
But then that that also means that within a similar triangle you have
Proportional sides
Okay, yeah.
which is useful.
Yeah, so yeah, I think that
being able to link them by a factor of 1/2. I guess that answers the questions. I don't know if my
1876 geometry tutor would be very happy with just my verbal reasoning but yeah
That's sort of the approach that I would take, so yeah.
Looks good to me
I'd let you in
Thank you
So, I'm gonna take over here for Toby and I'm gonna do number 5, which
involves defining the frustum of a pyramid
So I'm gonna go ahead an admit that I've already failed this test because I had no idea
What a frustum was. I had to look that up. Did you know what a frustum was?
No I didn't, so say we've had to
Look up some things here.
But it turns out a frustum, for all of you who are along with us
maybe some of you knew what it was, but if you have a pyramid,
and you cut the pyramid
partway
then this sort of
That goes all the way down to the bottom.
It's this part
- below the cut. - It's the part that's left over after you cut the tip off.
Yeah, if we remove the tip, what you've got left is this kind of
I'm pretty impressed with my drawings. Okay,
So what you've got left is like this sort of section of a pyramid
The lower section, like you cut off a smaller pyramid and removed it and then you've got this left.
This is me, defining
frustum
Yeah, you get points for that it seems because that's just the first part of the question, literally define it.
Define a frustum of a regular pyramid. Done.
Fist bump myself. And then
Are you gonna fist bump me?
I'm used to filming YouTube videos by myself. This is amazing.
There's somebody else to fist bump me. Then define lateral area
also had to look that up but
Lateral area is the area of these these sides the slanted side
so then there's obviously two more on the other side, but I can't draw these two right here and here so
Lateral area. Done, check, okay, and then prove the next part actually tells you to prove that's very exciting.
How nice of them.
Prove that the lateral area of a frustum of a regular pyramid is equal to half
the sum of the perimeters of the upper and lower bases.
So this is the upper base perimeter and then this is the lower base perimeter
Okay. Now we're gonna prove
multiplied by the slant height and then this slant height is this. This we'll do H
For the sake of hanging out with an Australian I'm gonna call it a trapezoid, but we would call a trapezoid
So these are all trapezoids
on the sides and to do the area of the of these you would
You could imagine that you take this little triangle from this side and move it over
To right there
multiply this
height by this base
And you get the area of the trapezoid which turns out to be
This distance here. It turns out to be the average between this distance this lower
part, the lower side of the trapezoid and the top part.
So we're gonna call this. I don't know. We'll call this like the base and the top.
So
To find the average of those you do base plus top over two
And then I want to find the area of this whole thing and that would be H
So the area of a trapezoid, I mean the formula for it ends up being the height
Sorry, not this-- I meant to refer to this height with my H. Now we're on the right track
This height is H now. We've got the formula for the area of a trapezoid, which is just the height of the trapezoid
times the base
length, which is this whole length here plus the top length over two. Now,
if we sum up all of those all four of them
Move this over for a second
So we've got all of the top links summed and all the bottom length summed over two
times H. I'm going to look back at what exactly I was proving which is
prove
The sum of the perimeters of the upper and lower bases multiplied by the slant height, half the sum. Cool
well, I've got my half, now I've got all of my
All of my top sides and all the bases added up but this right here is the top
perimeter. Woohoo
These four. There's the top perimeter.
And then these four here are the bottom perimeter.
So we've got that over two times the slant height and
that's what we were supposed to prove, right?
Yeah, I think you've got it.
Yaay.
That looks pretty good.
There it is.
Done. Fist bump.
You've been admitted to
MIT for your second degree.
After having to cheat and look up a frustum, yeah.
Nah I think we got there in the end. So I guess
Yeah, we've seen what these answers sort of look like, you've seen us sort of struggle with vocab here, but
yeah, I mean guys feel free to try some of these questions at home. It's not dangerous
And thanks Diana for joining me in this video
Thanks for having me. It was a lot of fun
Yeah, so we're gonna have a look at another MIT exam over on Diana's channel
So you guys can go and check out that if you want to.
We're going to look at
Physics, the first physics class I took which was called 8012 which is physics for masochists.
And this is the final.
Very shiny and beautiful and terrifying. Yeah. Alright. Thanks for watching