Main content

## Algebra basics

### Course: Algebra basics > Unit 8

Lesson 3: Pythagorean theorem# Intro to the Pythagorean theorem 2

Sal introduces the famous and super important Pythagorean theorem! Created by Sal Khan and CK-12 Foundation.

## Want to join the conversation?

- What are some practical uses for the Pythagorean Theorem?

I am curious because I find it very interesting and I want to know how I can use it.(119 votes)- How long of a ladder do you need to climb a 10 foot wall when you have to place the base of the ladder 5 feet from the wall? Square Root(10^2 + 5^2)...(107 votes)

- If I have 16+64="c"2 or squared how do I square root it? I mean square root 80.(167 votes)
- You don't even have to simplify the square root of 80. Just right it as √80 and you are done! This is because 80 doesn't have a perfect square root. It has a decimal, with an infinite amount of numbers after it.(7 votes)

- When you simplify the radical what is the Radical? I'm so confused some please help!(32 votes)
- try the course on squares, square roots, and radicals(3 votes)

- I'm so confused with the pratice problems. I'm usually really good in this topic, but since when do you put the AB and AC stuff together? I Don't know if I'm making any sense though.(16 votes)
- I think I get whet you're saying. . The theorem says A squared plus B squared equals C squared, so if you find a and b you can find C. The same goes for finding AC, AB, BC, or any other combo. if you find the other two, you can find the answer by turning i tinto an equation.(13 votes)

- how can I use pythagorean Theorem in my daily life. I am curious to know about it(7 votes)
- Pythagorean theorem is used in every case of our life....if u know the base and height of a tower....u can know the distance between that and u.....u can also find out distance between points in co-ordinate geometry.....(8 votes)

- At9:45, Sal says in the 2nd right triangle a=28 which is bigger than 21 but the 21 side is bigger?(4 votes)
- Sachin,

The drawing was 'not to scale'.

Even though he said it was a triangle that "looks like this", when he drew it, he only knew the hypotunuse and he knew one side, but he did not know until he did the calculation whether the "a" side was larger or smaller than 21.

So he just drew any right triangle and labeled the sides with the known information.

As it turned out, the triangle he drew did not truly represent the angles and lengths of a right triangle that has a hypotunuse of 35 and one length of 21.

I hope that helps.(7 votes)

- for the third (final) practice problem, at the11:00minute marker, b = the square root of 24^2 - 12^2. Since both terms under the square root are raised to the same power, I tried to subtract 24^2 from 12^2 and end up with b = 12. This is different than the final answer, and was just curious as to why this doesn't work...(4 votes)
- One of the fundamental rules in mathematics is to be sure and follow "the order of operations". This order is frequently referred to as PEMDAS (which is a pneumonic)

1. Parenthesis (or any grouping symbols such as brackets or braces) do what is inside the parenthesis first.

2. Exponents

3. Multiplication and Division, do in order from left to right

4. Addition and Subtraction, do in order from left to right(4 votes)

- What are Pythagorean triples?(4 votes)
- ""Pythagorean triples" are integer solutions to the Pythagorean Theorem, a2 + b2 = c2. I like "triplets," but "triples" seems to be the favored term. For a right triangle, the c side is the hypotenuse, the side opposite the right angle."

Example:

3^2 + 4^2 = 5^2(4 votes)

- In terms of "acceptable formats" for problem solving of Pythagorean Theorem is it possible to allow answers rounded off to two decimal places? This would allow the students the opportunity to practice with their calculators too.(5 votes)
- how do you know which side is the hypotenuse?(1 vote)
- The side opposite the right angle is the hypotenuse. It's also the longest side of the right triangle.(6 votes)

## Video transcript

Let's now talk about what is
easily one of the most famous theorems in all of
mathematics. And that's the Pythagorean
theorem. And it deals with
right triangles. So a right triangle is a
triangle that has a 90 degree angle in it. So the way I drew it
right here, this is our 90 degree angle. If you've never seen a 90 degree
angle before, the way to think about it is, if this
side goes straight left to right, this side goes straight
up and down. These sides are perpendicular,
or the angle between them is 90 degrees, or it is
a right angle. And the Pythagorean theorem
tells us that if we're dealing with a right triangle-- let me
write that down-- if we're dealing with a right triangle--
not a wrong triangle-- if we're dealing with
a right triangle, which is a triangle that has a right
angle, or a 90 degree angle in it, then the relationship
between their sides is this. So this side is a, this side
is b, and this side is c. And remember, the c that we're
dealing with right here is the side opposite the
90 degree angle. It's important to keep track
of which side is which. The Pythagorean theorem tells us
that if and only if this is a right triangle, then a squared
plus b squared is going to be equal
to c squared. And we can use this
information. If we know two of these, we
can then use this theorem, this formula to solve
for the third. And I'll give you one more piece
of terminology here. This long side, the side that
is the longest side of our right triangle, the side that
is opposite of our right angle, this right here-- it's
c in this example-- this is called a hypotenuse. A very fancy word for
a very simple idea. The longest side of a right
triangle, the side that is opposite the 90 degree angle,
is called the hypotentuse. Now that we know the Pythagorean
theorem, let's actually use it. Because it's one thing to know
something, but it's a lot more fun to use it. So let's say I have the
following right triangle. Let me draw it a little
bit neater than that. It's a right triangle. This side over here
has length 9. This side over here
has length 7. And my question is, what
is this side over here? Maybe we can call that--
we'll call that c. Well, c, in this case, once
again, it is the hypotenuse. It is the longest side. So we know that the sum of the
squares of the other side is going to be equal
to c squared. So by the Pythagorean theorem,
9 squared plus 7 squared is going to be equal
to c squared. 9 squared is 81, plus
7 squared is 49. 80 plus 40 is 120. Then we're going to have the 1
plus the 9, that's another 10, so this is going to
be equal to 130. So let me write it this way. The left-hand side is going to
be equal to 130, and that is equal to c squared. So what's c going
to be equal to? Let me rewrite it over here. c squared is equal to 130, or
we could say that c is equal to the square root of 130. And notice, I'm only taking
the principal root here, because c has to be positive. We're dealing with a distance,
so we can't take the negative square root. So we'll only take
the principal square root right here. And if we want to simplify this
a little bit, we know how to simplify our radicals. 130 is 2 times 65, which
is 5 times 13. Well, these are all prime
numbers, so that's about as simple as I can get.
c is equal to the square root of 130. Let's do another one of these. Maybe I want to keep this
Pythagorean theorem right there, just so we always
remember what we're referring to. So let's say I have a triangle
that looks like this. Let's see. Let's say it looks like that. And this is the right
angle, up here. Let's say that this side,
I'm going to call it a. The side, it's going
to have length 21. And this side right here is
going to be of length 35. So your instinct to solve for a,
might say, hey, 21 squared plus 35 squared is going to
be equal to a squared. But notice, in this situation,
35 is a hypotenuse. 35 is our c. It's the longest side of
our right triangle. So what the Pythagorean theorem
tells us is that a squared plus the other
non-longest side-- the other non-hypotenuse squared-- so a
squared plus 21 squared is going to be equal
to 35 squared. You always have to remember, the
c squared right here, the c that we're talking about, is
always going to be the longest side of your right triangle. The side that is opposite
of our right angle. This is the side that's opposite
of the right angle. So a squared plus 21 squared
is equal to 35 squared. And what do we have here? So 21 squared-- I'm tempted to
use a calculator, but I won't. So 21 times 21: 1 times 21
is 21, 2 times 21 is 42. It is 441. 35 squared. Once again, I'm tempted to use
a calculator, but I won't. 35 times 35: 5 times 5 is 25. Carry the 2. 5 times 3 is 15, plus 2 is 17. Put a 0 here, get rid
of that thing. 3 times 5 is 15. 3 times 3 is 9, plus 1 is 10. So it is 11-- let me do it in
order-- 5 plus 0 is 5, 7 plus 5 is 12, 1 plus 1 is 2,
bring down the 1. 1225. So this tells us that a squared
plus 441 is going to be equal to 35 squared,
which is 1225. Now, we could subtract
441 from both sides of this equation. The left-hand side just
becomes a squared. The right-hand side,
what do we get? We get 5 minus 1 is 4. We want to-- let me write this
a little bit neater here. Minus 441. So the left-hand side, once
again, they cancel out. a squared is equal to-- and then
on the right-hand side, what do we have to do? That's larger than that, but 2
is not larger than 4, so we're going to have to borrow. So that becomes a 12, or
regrouped, depending on how you want to view it. That becomes a 1. 1 is not greater than
4, so we're going to have to borrow again. Get rid of that. And then this becomes an 11. 5 minus 1 is 4. 12 minus 4 is 8. 11 minus 4 is 7. So a squared is equal to 784. And we could write, then,
that a is equal to the square root of 784. And once again, I'm very tempted
to use a calculator, but let's, well, let's not. Let's not use it. So this is 2 times, what? 392. And then this-- 390 times
2 is 78, yeah. And then this is
2 times, what? This is 2 times 196. That's right. 190 times 2 is-- yeah,
that's 2 times 196. 196 is 2 times-- I want to
make sure I don't make a careless mistake. 196 is 2 times 98. Let's keep going down here. 98 is 2 times 49. And, of course, we know
what that is. So notice, we have 2 times
2, times 2, times 2. So this is 2 to the
fourth power. So it's 16 times 49. So a is equal to the square
root of 16 times 49. I picked those numbers because
they're both perfect squares. So this is equal to the square
root of 16 is 4, times the square root of 49 is 7. It's equal to 28. So this side right here is going
to be equal to 28, by the Pythagorean theorem. Let's do one more of these. Can never get enough practice. So let's say I have
another triangle. I'll draw this one big. There you go. That's my triangle. That is the right angle. This side is 24. This side is 12. We'll call this side
right here b. Now, once again, always identify
the hypotenuse. That's the longest side,
the side opposite the 90 degree angle. You might say, hey, I don't know
that's the longest side. I don't know what b is yet. How do I know this is longest? And there, in that situation,
you say, well, it's the side opposite the 90 degree angle. So if that's the hypotenuse,
then this squared plus that squared is going to be
equal to 24 squared. So the Pythagorean theorem-- b
squared plus 12 squared is equal to 24 squared. Or we could subtract 12 squared
from both sides. We say, b squared is equal to
24 squared minus 12 squared, which we know is 144, and that
b is equal to the square root of 24 squared minus
12 squared. Now I'm tempted to use a
calculator, and I'll give into the temptation. So let's do it. The last one was so painful,
I'm still recovering. So 24 squared minus 12 squared
is equal to 24.78. So this actually turns into--
let me do it without a-- well, I'll do it halfway. 24 squared minus 12 squared
is equal to 432. So b is equal to the
square root of 432. And let's factor this again. We saw what the answer is, but
maybe we can write it in kind of a simplified radical form. So this is 2 times 216. 216, I believe, is
a-- let me see. I believe that's a
perfect square. So let me take the square
root of 216. Nope, not a perfect square. So 216, let's just keep going. 216 is 2 times 108. 108 is, we could say,
4 times what? 25 plus another 2-- 4 times
27, which is 9 times 3. So what do we have here? We have 2 times 2, times 4, so
this right here is a 16. 16 times 9 times 3. Is that right? I'm using a different
calculator. 16 times 9 times 3
is equal to 432. So this is going to be equal
to-- b is equal to the square root of 16 times 9, times 3,
which is equal to the square root of 16, which is 4 times the
square root of 9, which is 3, times the square root
of 3, which is equal to 12 roots of 3. So b is 12 times the
square root of 3. Hopefully you found
that useful.