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In the last video, we
showed that the ratios

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of the sides of a
30-60-90 triangle

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are-- if we assume
the longest side is

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x, if the hypotenuse is x.

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Then the shortest side is
x/2 and the side in between,

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the side that's opposite
the 60 degree side,

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is square root of 3x/2.

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Or another way to think about it
is if the shortest side is 1--

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Now I'll do the shortest side,
then the medium size, then

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the longest side.

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So if the side opposite
the 30 degree side is 1,

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then the side opposite
the 60 degree side

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is square root of 3 times that.

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So it's going to be
square root of 3.

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And then the hypotenuse
is going to be twice that.

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In the last video,
we started with x

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and we said that the
30 degree side is x/2.

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But if the 30 degree
side is 1, then this

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is going to be twice that.

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So it's going to be 2.

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This right here is the side
opposite the 30 degree side,

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opposite the 60 degree side,
and then the hypotenuse opposite

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the 90 degree side.

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And so, in general, if
you see any triangle that

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has those ratios, you say hey,
that's a 30-60-90 triangle.

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Or if you see a
triangle that you

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know is a 30-60-90 triangle,
you could say, hey,

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I know how to figure out
one of the sides based

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on this ratio right over here.

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Just an example, if
you see a triangle that

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looks like this, where the
sides are 2, 2 square root of 3,

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and 4.

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Once again, the ratio of
2 to 2 square root of 3

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is 1 to square root of 3.

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The ratio of 2 to 4 is
the same thing as 1 to 2.

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This right here must
be a 30-60-90 triangle.

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What I want to introduce
you to in this video

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is another important
type of triangle

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that shows up a lot in geometry
and a lot in trigonometry.

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And this is a 45-45-90 triangle.

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Or another way to
think about is if I

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have a right triangle
that is also isosceles.

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You obviously can't have a right
triangle that is equilateral,

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because an equilateral triangle
has all of their angles

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have to be 60 degrees.

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But you can have
a right angle, you

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can have a right triangle,
that is isosceles.

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And isosceles--
let me write this--

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this is a right
isosceles triangle.

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And if it's isosceles,
that means two of the sides

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are equal.

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So these are the two
sides that are equal.

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And then if the two
sides are equal,

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we have proved to ourselves
that the base angles are equal.

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And if we called the measure
of these base angles x,

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then we know that x plus x plus
90 have to be equal to 180.

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Or if we subtract
90 from both sides,

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you get x plus x is equal
to 90 or 2x is equal to 90.

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Or if you divide
both sides by 2,

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you get x is equal
to 45 degrees.

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So a right isosceles
triangle can also be called--

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and this is the more
typical name for it--

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it can also be called
a 45-45-90 triangle.

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And what I want to do
this video is come up

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with the ratios for the
sides of a 45-45-90 triangle,

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just like we did for
a 30-60-90 triangle.

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And this one's actually
more straightforward.

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Because in a 45-45-90 triangle,
if we call one of the legs x,

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the other leg is
also going to be x.

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And then we can use
the Pythagorean Theorem

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to figure out the length
of the hypotenuse.

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So the length of the
hypotenuse, let's call that c.

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So we get x squared
plus x squared.

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That's the square of
length of both of the legs.

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So when we sum those
up, that's going

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to have to be
equal to c squared.

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This is just straight out
of the Pythagorean theorem.

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So we get 2x squared
is equal to c squared.

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We can take the principal
root of both sides of that.

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I wanted to just
change it to yellow.

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Last, take the principal
root of both sides of that.

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The left-hand side you
get, principal root of 2

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is just square
root of 2, and then

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the principal root of x
squared is just going to be x.

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So you're going to have x
times the square root of 2

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is equal to c.

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So if you have a right isosceles
triangle, whatever the two

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legs are, they're going
to have the same length.

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That's why it's isosceles.

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The hypotenuse is going to be
square root of 2 times that.

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So c is equal to x times
the square root of 2.

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So for example, if you have a
triangle that looks like this.

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Let me draw it a
slightly different way.

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It's good to have to orient
ourselves in different ways

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every time.

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So if we see a triangle
that's 90 degrees,

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45 and 45 like this,
and you really just

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have to know two of
these angles to know

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what the other one
is going to be,

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and if I tell you that
this side right over here

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is 3-- I actually don't
even have to tell you

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that this other
side's going to be 3.

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This is an isosceles
triangle, so those two legs

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are going to be the same.

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And you won't even have to
apply the Pythagorean theorem

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if you know this--
and this is a good one

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to know-- that the hypotenuse
here, the side opposite the 90

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degree side, is just going
to be square root of 2

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times the length of
either of the legs.

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So it's going to be 3
times the square root of 2.

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So the ratio of the
size of the hypotenuse

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in a 45-45-90 triangle or
a right isosceles triangle,

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the ratio of the sides are
one of the legs can be 1.

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Then the other leg is going
to have the same measure,

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the same length, and then
the hypotenuse is going

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to be square root of 2
times either of those.

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1 to 1, 2 square root of 2.

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So this is 45-45-90.

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That's the ratios.

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And just as a review,
if you have a 30-60-90,

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the ratios were 1 to
square root of 3 to 2.

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And now we'll apply this
in a bunch of problems.