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### Course: High school geometry > Unit 5

Lesson 3: Special right triangles# Special right triangles proof (part 2)

Showing the ratios of the sides of a 45-45-90 triangle are 1:1:sqrt(2). Created by Sal Khan.

## Want to join the conversation?

- Can you end up with anything other than a isosceles triangle if you have one 45 degree angle and one 90 degree angle?(17 votes)
- Nope, because a triangle
*ALWAYS*has 180˚. No more, no less. If two angles are 45 and 90, then the other HAS to be 45 to make it add to 180. Hope this helps!(38 votes)

- does anyone know where hypotenuse came from?(14 votes)
- The word hypotenuse means essentially "length under", and derives from Latin hypotēnūsa (according to Wikipedia: http://en.wikipedia.org/wiki/Hypotenuse)(20 votes)

- I understand how the 45-45-90 triangle works, but does anyone know a 45-45-90 triangle with whole numbers?

(i need three whole numbers, kind of like the 3-4-5 triangle, but i need it in a 45-45-90 triangle)

Thanks!(5 votes)- Since a 45-45-90 multiplies by √2 to get from hypotenuse to side, there is no integer representation.(22 votes)

- If the two legs are x and the hypotenuse is three square root 2 how do I find x?(8 votes)
- Well, it depends on what information you have and what you are trying to find(3 votes)

- how do you do pythagorean theorem please(3 votes)
- It was theorem proposed by Pythagoras,
**which deals with Right angled Triangles only**

Pythagorean Theorem just states that in any Right Triangle(With a 90 degree angle) the Length of Hypotenuse squared (Side opposite to 90 degree) is equal to the Sum of the length of squares of its base and adjacent side.`Commonly known as A^2 + B^2 = C^2`

*To be clear, the Side CB in Triangle ABC is equal to the root of Sum of the other two sides, that is √( AB+AC)***C Angle CAB is a 90 degree angle**

• Which Makes CB Hypotenuse

• • By Pythagorean Theorem CB^2 = AB^2 + AC^2

• • CB = √AB^2 + BC^2

• • Try doing a problem, its on KA, and is Fun!

• •

• •

• •

A•--------------------------• B**An Example**

Let a Triangle have Legs(Base & Adjacent) of 12 and 5cm. So, the The Third side or Hypotenuse is 12^2 + 5^2 = > 144+25 = √169 = 13cm`For a reference see Sal's beautiful playlist on Pythagorean Theorem :- https://www.khanacademy.org/math/geometry/right_triangles_topic/pyth_theor/v/pythagorean-theorem`

(8 votes)

- In a 45-45-90 triangle, how would you solve for the other two lengths if the base is 8*square root 2? Or please give me an example, thank-you.

I couldn't watch the video because of computer difficulties, my apologies if I'm a little off topic.(5 votes)- Did you mean the hypotenuse was 8*sqrt(2) units long?

If you know the hypotenuse's length of a 45-45-90 triangle, divide the hypotenuse's length by sqrt(2) to find the length of both of the legs. In this case, the legs have a length of 8 units.

Did you mean one of the legs was 8*sqrt(2) units long?

If you know the length of one of the legs of a 45-45-90 triangle, the other leg has the same length. For this triangle, the other leg has a length of 8*sqrt(2) units. The hypotenuse's length can be found by multiplying the leg's length by sqrt(2). Tis triangle's hypotenuse has a length of 16 units.

In a 45-45-90 triangle, both of the legs have the same length and the ratio of one leg to the hypotenuse is 1:sqrt(2). I hope this helps!(3 votes)

- This may sound like a stupid question but what video precedes this one? Sal says in the previous video... but i cant find it.(2 votes)
- I think this is the video you are looking for: http://www.khanacademy.org/math/geometry/triangles/v/30-60-90-triangle-side-ratios-proof(6 votes)

- Where did the name right angle come from?(4 votes)
- The meaning of 'right' in right angle possibly refers to the Latin adjective rectus erect, straight, upright, perpendicular'.

In this case the origin of the term could come from architecture where right angles would be used in building structures. In this scenario right would refer to the correct way of doing something such as laying beams for a house.(1 vote)

- sal, how do write so neat with a mousepad? when i write, it is terrible.(3 votes)
- There are USB drawing tablets. It's like using a stylus on a mouse pad.

Hope that helps!(2 votes)

- How did they discover the special right triangles?(3 votes)
- That's a good question. Simply put, with any given set of angles, there is one side ratio. Now, for most general cases, you either don't know what the angles are, or they're something like 63-27-90, in which case it is better to use the normal Pythagorean Theorem, since you don't want to make a new formula every time you run into a new angle. But for cases like 30-60-90, or 45-45-90, it is worth your time to figure out a side formula and then apply it every time you need to. So, really, there aren't special right triangles. You can apply this method to any triangle. But since people use this technique primarily on 30-60-90 and 45-45-90 triangles, they're called 'special'.(0 votes)

## Video transcript

In the last video, we
showed that the ratios of the sides of a
30-60-90 triangle are-- if we assume
the longest side is x, if the hypotenuse is x. Then the shortest side is
x/2 and the side in between, the side that's opposite
the 60 degree side, is square root of 3x/2. Or another way to think about it
is if the shortest side is 1-- Now I'll do the shortest side,
then the medium size, then the longest side. So if the side opposite
the 30 degree side is 1, then the side opposite
the 60 degree side is square root of 3 times that. So it's going to be
square root of 3. And then the hypotenuse
is going to be twice that. In the last video,
we started with x and we said that the
30 degree side is x/2. But if the 30 degree
side is 1, then this is going to be twice that. So it's going to be 2. This right here is the side
opposite the 30 degree side, opposite the 60 degree side,
and then the hypotenuse opposite the 90 degree side. And so, in general, if
you see any triangle that has those ratios, you say hey,
that's a 30-60-90 triangle. Or if you see a
triangle that you know is a 30-60-90 triangle,
you could say, hey, I know how to figure out
one of the sides based on this ratio right over here. Just an example, if
you see a triangle that looks like this, where the
sides are 2, 2 square root of 3, and 4. Once again, the ratio of
2 to 2 square root of 3 is 1 to square root of 3. The ratio of 2 to 4 is
the same thing as 1 to 2. This right here must
be a 30-60-90 triangle. What I want to introduce
you to in this video is another important
type of triangle that shows up a lot in geometry
and a lot in trigonometry. And this is a 45-45-90 triangle. Or another way to
think about is if I have a right triangle
that is also isosceles. You obviously can't have a right
triangle that is equilateral, because an equilateral triangle
has all of their angles have to be 60 degrees. But you can have
a right angle, you can have a right triangle,
that is isosceles. And isosceles--
let me write this-- this is a right
isosceles triangle. And if it's isosceles,
that means two of the sides are equal. So these are the two
sides that are equal. And then if the two
sides are equal, we have proved to ourselves
that the base angles are equal. And if we called the measure
of these base angles x, then we know that x plus x plus
90 have to be equal to 180. Or if we subtract
90 from both sides, you get x plus x is equal
to 90 or 2x is equal to 90. Or if you divide
both sides by 2, you get x is equal
to 45 degrees. So a right isosceles
triangle can also be called-- and this is the more
typical name for it-- it can also be called
a 45-45-90 triangle. And what I want to do
this video is come up with the ratios for the
sides of a 45-45-90 triangle, just like we did for
a 30-60-90 triangle. And this one's actually
more straightforward. Because in a 45-45-90 triangle,
if we call one of the legs x, the other leg is
also going to be x. And then we can use
the Pythagorean Theorem to figure out the length
of the hypotenuse. So the length of the
hypotenuse, let's call that c. So we get x squared
plus x squared. That's the square of
length of both of the legs. So when we sum those
up, that's going to have to be
equal to c squared. This is just straight out
of the Pythagorean theorem. So we get 2x squared
is equal to c squared. We can take the principal
root of both sides of that. I wanted to just
change it to yellow. Last, take the principal
root of both sides of that. The left-hand side you
get, principal root of 2 is just square
root of 2, and then the principal root of x
squared is just going to be x. So you're going to have x
times the square root of 2 is equal to c. So if you have a right isosceles
triangle, whatever the two legs are, they're going
to have the same length. That's why it's isosceles. The hypotenuse is going to be
square root of 2 times that. So c is equal to x times
the square root of 2. So for example, if you have a
triangle that looks like this. Let me draw it a
slightly different way. It's good to have to orient
ourselves in different ways every time. So if we see a triangle
that's 90 degrees, 45 and 45 like this,
and you really just have to know two of
these angles to know what the other one
is going to be, and if I tell you that
this side right over here is 3-- I actually don't
even have to tell you that this other
side's going to be 3. This is an isosceles
triangle, so those two legs are going to be the same. And you won't even have to
apply the Pythagorean theorem if you know this--
and this is a good one to know-- that the hypotenuse
here, the side opposite the 90 degree side, is just going
to be square root of 2 times the length of
either of the legs. So it's going to be 3
times the square root of 2. So the ratio of the
size of the hypotenuse in a 45-45-90 triangle or
a right isosceles triangle, the ratio of the sides are
one of the legs can be 1. Then the other leg is going
to have the same measure, the same length, and then
the hypotenuse is going to be square root of 2
times either of those. 1 to 1, 2 square root of 2. So this is 45-45-90. That's the ratios. And just as a review,
if you have a 30-60-90, the ratios were 1 to
square root of 3 to 2. And now we'll apply this
in a bunch of problems.