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## Mathematics II (2018 edition)

### Unit 10: Lesson 1

Pythagorean theorem- Intro to the Pythagorean theorem
- Intro to the Pythagorean theorem 2
- Pythagorean theorem word problem: fishing boat
- Pythagorean theorem example
- Pythagorean theorem word problem: carpet
- Use Pythagorean theorem to find right triangle side lengths
- Pythagorean theorem challenge
- Introduction to the Pythagorean theorem
- Pythagorean theorem II

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# Pythagorean theorem II

More Pythagorean Theorem examples. Introduction to 45-45-90 triangles. Created by Sal Khan.

## Video transcript

I promised you that I'd give
you some more Pythagorean theorem problems, so I will
now give you more Pythagorean theorem problems. And once again, this is
all about practice. Let's say I had a triangle--
that's an ugly looking right triangle, let me draw another
one --and if I were to tell you that that side is 7, the
side is 6, and I want to figure out this side. Well, we learned in the last
presentation: which of these sides is the hypotenuse? Well, here's the right angle,
so the side opposite the right angle is the hypotenuse. So what we want to do
is actually figure out the hypotenuse. So we know that 6 squared
plus 7 squared is equal to the hypotenuse squared. And in the Pythagorean theorem
they use C to represent the hypotenuse, so we'll
use C here as well. And 36 plus 49 is
equal to C squared. 85 is equal to C squared. Or C is equal to the
square root of 85. And this is the part that most
people have trouble with, is actually simplifying
the radical. So the square root of 85: can I
factor 85 so it's a product of a perfect square and
another number? 85 isn't divisible by 4. So it won't be divisible by 16
or any of the multiples of 4. 5 goes into 85 how many times? No, that's not perfect
square, either. I don't think 85 can be
factored further as a product of a perfect
square and another number. So you might correct
me; I might be wrong. This might be good exercise for
you to do later, but as far as I can tell we have
gotten our answer. The answer here is the
square root of 85. And if we actually wanted to
estimate what that is, let's think about it: the square root
of 81 is 9, and the square root of 100 is 10 , so it's some
place in between 9 and 10, and it's probably a little
bit closer to 9. So it's 9 point something,
something, something. And that's a good reality
check; that makes sense. If this side is 6, this side
is 7, 9 point something, something, something makes
sense for that length. Let me give you
another problem. [DRAWING] Let's say that this is 10 . This is 3. What is this side? First, let's identify
our hypotenuse. We have our right angle here,
so the side opposite the right angle is the hypotenuse and
it's also the longest side. So it's 10. So 10 squared is equal to
the sum of the squares of the other two sides. This is equal to 3 squared--
let's call this A. Pick it arbitrarily. --plus A squared. Well, this is 100, is equal to
9 plus A squared, or A squared is equal to 100 minus 9. A squared is equal to 91. I don't think that can be
simplified further, either. 3 doesn't go into it. I wonder, is 91 a prime number? I'm not sure. As far as I know, we're
done with this problem. Let me give you another
problem, And actually, this time I'm going to include one
extra step just to confuse you because I think you're getting
this a little bit too easily. Let's say I have a triangle. And once again, we're dealing
all with right triangles now. And never are you going to
attempt to use the Pythagorean theorem unless you know for a
fact that's all right triangle. But this example, we know
that this is right triangle. If I would tell you the length
of this side is 5, and if our tell you that this angle is 45
degrees, can we figure out the other two sides of
this triangle? Well, we can't use the
Pythagorean theorem directly because the Pythagorean theorem
tells us that if have a right triangle and we know two of the
sides that we can figure out the third side. Here we have a right
triangle and we only know one of the sides. So we can't figure out
the other two just yet. But maybe we can use this extra
information right here, this 45 degrees, to figure out another
side, and then we'd be able use the Pythagorean theorem. Well, we know that the
angles in a triangle add up to 180 degrees. Well, hopefully you know
the angles in a triangle add up to 180 degrees. If you don't it's my fault
because I haven't taught you that already. So let's figure out what
the angles of this triangle add up to. Well, I mean we know they add
up to 180, but using that information, we could figure
out what this angle is. Because we know that this angle
is 90, this angle is 45. So we say 45-- lets call this
angle x; I'm trying to make it messy --45 plus 90--
this [INAUDIBLE] is a 90 degree angle --plus
is equal to 180 degrees. And that's because the
angles in a triangle always add up to 180 degrees. So if we just solve for x, we
get 135 plus x is equal to 180. Subtract 135 from both sides. We get x is equal
to 45 degrees. Interesting. x is also 45 degrees. So we have a 90 degree angle
and two 45 degree angles. Now I'm going to give you
another theorem that's not named after the head
of a religion or the founder of religion. I actually don't think this
theorem doesn't have a name at. All It's the fact that if I
have another triangle --I'm going to draw another triangle
out here --where two of the base angles are the same-- and
when I say base angle, I just mean if these two angles are
the same, let's call it a. They're both a --then the sides
that they don't share-- these angles share this side, right? --but if we look at the sides
that they don't share, we know that these sides are equal. I forgot what we call
this in geometry class. Maybe I'll look it up in
another presentation; I'll let you know. But I got this far without
knowing what the name of the theorem is. And it makes sense; you don't
even need me to tell you that. If I were to change one of
these angles, the length would also change. Or another way to think about
it, the only way-- no, I don't confuse you too much. But you can visually see that
if these two sides are the same, then these two angles
are going to be the same. If you changed one of these
sides' lengths, then the angles will also change, or the angles
will not be equal anymore. But I'll leave that for
you to think about. But just take my word for it
right now that if two angles in a triangle are equivalent, then
the sides that they don't share are also equal in length. Make sure you remember: not the
side that they share-- because that can't be equal to anything
--it's the side that they don't share are equal in length. So here we have an example
where we have to equal angles. They're both 45 degrees. So that means that the sides
that they don't share-- this is the side they share, right? Both angle share this side --so
that means that the side that they don't share are equal. So this side is
equal to this side. And I think you might be
experiencing an ah-hah moment that right now. Well this side is equal to this
side-- I gave you at the beginning of this problem that
this side is equal to 5 --so then we know that this
side is equal to 5. And now we can do the
Pythagorean theorem. We know this is the
hypotenuse, right? So we can say 5 squared plus 5
squared is equal to-- let's say C squared, where C is the
length of the hypotenuse --5 squared plus 5 squared-- that's
just 50 --is equal to C squared. And then we get C is equal
to the square root of 50. And 50 is 2 times 25, so C is
equal to 5 square roots of 2. Interesting. So I think I might have given
you a lot of information there. If you get confused, maybe you
want to re-watch this video. But on the next video I'm
actually going to give you more information about this type of
triangle, which is actually a very common type of triangle
you'll see in geometry and trigonometry 45,
45, 90 triangle. And it makes sense why it's
called that because it has 45 degrees, 45 degrees,
and a 90 degree angle. And I'll actually show you
a quick way of using that information that it is a 45,
45, 90 degree triangle to figure out the size if you're
given even one of the sides. I hope I haven't confused you
too much, and I look forward to seeing you in the
next presentation. See you later.