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I promised you that I'd give
you some more Pythagorean

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theorem problems, so I will
now give you more Pythagorean

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theorem problems.

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And once again, this is
all about practice.

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Let's say I had a triangle--
that's an ugly looking right

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triangle, let me draw another
one --and if I were to tell

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you that that side is 7, the
side is 6, and I want to

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figure out this side.

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Well, we learned in the last
presentation: which of these

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sides is the hypotenuse?

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Well, here's the right angle,
so the side opposite the right

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angle is the hypotenuse.

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So what we want to do
is actually figure

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out the hypotenuse.

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So we know that 6 squared
plus 7 squared is equal to

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the hypotenuse squared.

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And in the Pythagorean theorem
they use C to represent the

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hypotenuse, so we'll
use C here as well.

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And 36 plus 49 is
equal to C squared.

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85 is equal to C squared.

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Or C is equal to the
square root of 85.

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And this is the part that most
people have trouble with, is

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actually simplifying
the radical.

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So the square root of 85: can I
factor 85 so it's a product of

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a perfect square and
another number?

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85 isn't divisible by 4.

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So it won't be divisible by 16
or any of the multiples of 4.

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5 goes into 85 how many times?

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No, that's not perfect
square, either.

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I don't think 85 can be
factored further as a

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product of a perfect
square and another number.

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So you might correct
me; I might be wrong.

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This might be good exercise for
you to do later, but as far as

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I can tell we have
gotten our answer.

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The answer here is the
square root of 85.

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And if we actually wanted to
estimate what that is, let's

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think about it: the square root
of 81 is 9, and the square root

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of 100 is 10 , so it's some
place in between 9 and 10, and

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it's probably a little
bit closer to 9.

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So it's 9 point something,
something, something.

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And that's a good reality
check; that makes sense.

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If this side is 6, this side
is 7, 9 point something,

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something, something makes
sense for that length.

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Let me give you
another problem.

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[DRAWING]

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Let's say that this is 10 .

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This is 3.

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What is this side?

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First, let's identify
our hypotenuse.

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We have our right angle here,
so the side opposite the right

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angle is the hypotenuse and
it's also the longest side.

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So it's 10.

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So 10 squared is equal to
the sum of the squares

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of the other two sides.

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This is equal to 3 squared--
let's call this A.

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Pick it arbitrarily.

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--plus A squared.

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Well, this is 100, is equal to
9 plus A squared, or A squared

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is equal to 100 minus 9.

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A squared is equal to 91.

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I don't think that can be
simplified further, either.

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3 doesn't go into it.

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I wonder, is 91 a prime number?

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I'm not sure.

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As far as I know, we're
done with this problem.

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Let me give you another
problem, And actually, this

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time I'm going to include one
extra step just to confuse you

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because I think you're getting
this a little bit too easily.

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Let's say I have a triangle.

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And once again, we're dealing
all with right triangles now.

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And never are you going to
attempt to use the Pythagorean

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theorem unless you know for a
fact that's all right triangle.

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But this example, we know
that this is right triangle.

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If I would tell you the length
of this side is 5, and if our

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tell you that this angle is 45
degrees, can we figure out the

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other two sides of
this triangle?

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Well, we can't use the
Pythagorean theorem directly

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because the Pythagorean theorem
tells us that if have a right

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triangle and we know two of the
sides that we can figure

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out the third side.

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Here we have a right
triangle and we only

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know one of the sides.

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So we can't figure out
the other two just yet.

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But maybe we can use this extra
information right here, this 45

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degrees, to figure out another
side, and then we'd be able

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use the Pythagorean theorem.

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Well, we know that the
angles in a triangle

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add up to 180 degrees.

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Well, hopefully you know
the angles in a triangle

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add up to 180 degrees.

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If you don't it's my fault
because I haven't taught

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you that already.

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So let's figure out what
the angles of this

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triangle add up to.

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Well, I mean we know they add
up to 180, but using that

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information, we could figure
out what this angle is.

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Because we know that this angle
is 90, this angle is 45.

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So we say 45-- lets call this
angle x; I'm trying to make it

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messy --45 plus 90--
this just symbolizes

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a 90 degree angle --plus x
is equal to 180 degrees.

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And that's because the
angles in a triangle always

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add up to 180 degrees.

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So if we just solve for x, we
get 135 plus x is equal to 180.

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Subtract 135 from both sides.

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We get x is equal
to 45 degrees.

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Interesting.

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x is also 45 degrees.

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So we have a 90 degree angle
and two 45 degree angles.

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Now I'm going to give you
another theorem that's not

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named after the head
of a religion or the

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founder of religion.

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I actually don't think this
theorem doesn't have a name at.

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All It's the fact that if I
have another triangle --I'm

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going to draw another triangle
out here --where two of the

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base angles are the same-- and
when I say base angle, I just

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mean if these two angles are
the same, let's call it a.

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They're both a --then the sides
that they don't share-- these

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angles share this side, right?

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--but if we look at the sides
that they don't share, we know

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that these sides are equal.

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I forgot what we call
this in geometry class.

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Maybe I'll look it up in
another presentation;

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I'll let you know.

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But I got this far without
knowing what the name

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of the theorem is.

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And it makes sense; you don't
even need me to tell you that.

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If I were to change one of
these angles, the length

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would also change.

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Or another way to think about
it, the only way-- no, I

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don't confuse you too much.

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But you can visually see that
if these two sides are the

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same, then these two angles
are going to be the same.

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If you changed one of these
sides' lengths, then the angles

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will also change, or the angles
will not be equal anymore.

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But I'll leave that for
you to think about.

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But just take my word for it
right now that if two angles in

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a triangle are equivalent, then
the sides that they don't share

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are also equal in length.

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Make sure you remember: not the
side that they share-- because

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that can't be equal to anything
--it's the side that they don't

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share are equal in length.

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So here we have an example
where we have to equal angles.

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They're both 45 degrees.

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So that means that the sides
that they don't share-- this is

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the side they share, right?

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Both angle share this side --so
that means that the side that

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they don't share are equal.

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So this side is
equal to this side.

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And I think you might be
experiencing an ah-hah

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moment that right now.

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Well this side is equal to this
side-- I gave you at the

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beginning of this problem that
this side is equal to 5 --so

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then we know that this
side is equal to 5.

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And now we can do the
Pythagorean theorem.

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We know this is the
hypotenuse, right?

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So we can say 5 squared plus 5
squared is equal to-- let's say

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C squared, where C is the
length of the hypotenuse --5

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squared plus 5 squared-- that's
just 50 --is equal

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to C squared.

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And then we get C is equal
to the square root of 50.

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And 50 is 2 times 25, so C is
equal to 5 square roots of 2.

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Interesting.

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So I think I might have given
you a lot of information there.

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If you get confused, maybe you
want to re-watch this video.

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But on the next video I'm
actually going to give you more

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information about this type of
triangle, which is actually a

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very common type of triangle
you'll see in geometry and

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trigonometry 45,
45, 90 triangle.

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And it makes sense why it's
called that because it has

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45 degrees, 45 degrees,
and a 90 degree angle.

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And I'll actually show you
a quick way of using that

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information that it is a 45,
45, 90 degree triangle to

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figure out the size if you're
given even one of the sides.

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I hope I haven't confused you
too much, and I look forward

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to seeing you in the
next presentation.

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See you later.