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### Course: 8th grade (Illustrative Mathematics) > Unit 8

Lesson 6: Lesson 6: Finding side lengths of triangles# Intro to the Pythagorean theorem

The Pythagorean theorem is a cornerstone of math that helps us find the missing side length of a right triangle. In a right triangle with sides A, B, and hypotenuse C, the theorem states that A² + B² = C². The hypotenuse is the longest side, opposite the right angle. Created by Sal Khan.

## Want to join the conversation?

- close towards the end how did you solve the square root?(87 votes)
- you do the principal root:the square root of 2*54=the square root of 108 and the square root of 2*27=the square root of 54 and the square root of 3*9=the square root of 27 and the square root of 3*3=the square root of 9 and 2*2*3*3*3=108/the square root of 2*2*3*3*3=the square root of 108.(87 votes)

- It's a wonder how Pythagoras thought this whole thing up, he's a pure genius.(25 votes)
- It is now shown that this was known long before Pythagoras, he just got the credit for other people's work. Further, he did not really like the idea of irrational numbers which is a consequence of the theorem. Interesting article on this is at https://www.britannica.com/science/Pythagorean-theorem which also talks about his life and how he may have come into contact with those who already had applied the Theorem.(29 votes)

- how you know which one is A,B, or C?(10 votes)
- c is equal to the hypotenuse and a and b are the shorter sides (you can choose which one you want to be a or b)(32 votes)

- Alright, buckle up, because we're diving into the wonderful world of the Pythagorean theorem! Don't worry, I'm going to make it as fun and exciting as a roller coaster ride. 🎢

So, picture this: you're on a quest to uncover the secrets of right triangles. What's a right triangle, you ask? It's like a ninja triangle that has one angle that's super square – like the corners of your favorite chocolate bar. And guess what? The Pythagorean theorem is like the secret map that helps you unravel the mystery of their sides.

Let's meet our triangle heroes: the legs (let's call them Alex and Bella) and the hypotenuse (let's call it Charlie). Alex and Bella are the sides that stick around the right angle, and Charlie is the longest side that stretches across like a superhero cape.

Now, the Pythagorean theorem is like a fancy spell that connects Alex, Bella, and Charlie. It's written as "Alex squared plus Bella squared equals Charlie squared." Math wizards use letters to stand in for numbers, so "squared" just means you multiply a number by itself. 🧙♂️

Let's break down our secret spell:

"Alex" is one of the leg's lengths, and "Bella" is the other leg's length. Imagine they're talking to each other, going "Hey, I'm Alex!" and "Hi, I'm Bella!" Cute, right?

"Charlie" is the hypotenuse's length. Charlie's like, "I'm the cool slanty side over here!"

When we say "Alex squared," it's like saying "Alex times Alex." Same goes for "Bella squared" and "Charlie squared."

So, when we say "Alex squared plus Bella squared equals Charlie squared," we're saying "Alex times Alex plus Bella times Bella equals Charlie times Charlie." It's like a math party with letters dancing around!

Let's rock this with an example. Imagine Alex is 3 units long and Bella is 4 units long. Our goal? Figure out how long Charlie is. Ready?

First, we square Alex: 3² = 9. Imagine Alex's square dance moves!

Then, we square Bella: 4² = 16. Bella's got some smooth square moves too!

Now, we add Alex's square and Bella's square: 9 + 16 = 25. It's a party, and the dance floor is lit!

Finally, we need to know how long Charlie is, so we take the square root of 25: √25 = 5. Charlie's got some severe superhero vibes!

So, in our awesome triangle adventure, Charlie, the hypotenuse is five units long. Ta-da! 🎉

The Pythagorean theorem isn't just for math class; it's like a puzzle-solving tool that architects, engineers, and even treasure hunters use. Imagine building a treehouse with Alex and Bella, making sure Charlie's rope ladder is just the right length. Or imagine searching for hidden treasure, using the theorem to measure how far X marks the spot.

So, whether exploring ancient mysteries or creating your own superhero squad of triangles, remember that the Pythagorean theorem is your trusty sidekick. It's a dance of numbers that unlocks the secrets of right triangles, making math an epic adventure filled with discovery and a dash of superhero magic! 🦸♂️🔮🌟(33 votes)

- What is the Pythagorean theorem?I need help trying to understand it.(9 votes)
- The Pythagorean theorem consists of a formula a^2+b^2=c^2 which is used to figure out the value of (mostly) the hypotenuse in a right triangle. The a and b are the 2 "non-hypotenuse" sides of the triangle (Opposite and Adjacent). Once you progress, you will be given the hypotenuse and would be needed to find the opposite or the adjacent side (a or b). The Pythagorean theorem is a simple formula which uses the squared value of a and b; for example "a=3 and b=4, what is the value of c?" you square a (3^2=9=a) and b (4^2=16=b) and add the 2 values (9+16=25) to get to c. To complete the question, you have to square root c's value (square root of 25=5) because the formula says c^2 and not just c. Once you have done that, you can check your answer by squaring a,b and c to see if you have added and divided (Square-rooted) correctly.

Hope this helps!(40 votes)

- Hi, I have a question. What is the square root? In the video at5:27he said that in order to complete the equation you have to take the positive square root of both sides, which for 25 would equal 5. But what does that mean? How did he get 5 from 25? What did he do, what did he divide 25 by and why did he divide that and not another number? I will be waiting for a response thank you to those that reply, I will be very thankful because I know I would be taking time away from you just so you can answer my question. Thanks!(10 votes)
- A square root is a number that produces a specified quantity when multiplied by itself.

It goes hand in hand with exponents and squares. 2 squared is 4, and the square root of 4 is 2.

The square root is just the number that, when multiplied by itself, equals the original number you are starting with.

The square root of 25 is 5. Because 5 * 5 is equal to 25.

The square root of 49 is 7. Because 7 * 7 is 49.

The square root of 625 is 25. Because 25 * 25 is equal to 625.

The square root of 89,737,543 is 9473. (Approximately)

If you still have trouble with this concept:

https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:rational-exponents-radicals/x2f8bb11595b61c86:radicals/v/understanding-square-roots(19 votes)

- This doesn't have much to do with the video, but at5:28, Sal says we take the positive square root of both sides. Is there a negative square root?(6 votes)
- Yes, for example, the positive square root of 25 is 5 and the negative square root is -5. When you square negative numbers, you get a positive answer, therefore the square root of a positive number will have both a positive and a negative.(14 votes)

- how do you do this(6 votes)
- Watch the video. It explains it very well(6 votes)

- Couldn't you have just solved 6 squared + b squared = 12 squared using an equation? Or doing 12 squared minus 6 squared??(6 votes)
- Tell me if I'm wrong, but I think this is exactly what Sal does in the video. He explains the theorem and the formula, then applies it by taking a problem and turning it into an equation.(10 votes)

- How did you find the square root in this video? Also love this app, helps with my studies so well!(10 votes)

## Video transcript

In this video we're going
to get introduced to the Pythagorean theorem,
which is fun on its own. But you'll see as you learn
more and more mathematics it's one of those cornerstone
theorems of really all of math. It's useful in geometry,
it's kind of the backbone of trigonometry. You're also going to use
it to calculate distances between points. So it's a good thing to really
make sure we know well. So enough talk on my end. Let me tell you what the
Pythagorean theorem is. So if we have a triangle, and
the triangle has to be a right triangle, which means that one
of the three angles in the triangle have to be 90 degrees. And you specify that it's
90 degrees by drawing that little box right there. So that right there is-- let
me do this in a different color-- a 90 degree angle. Or, we could call
it a right angle. And a triangle that has
a right angle in it is called a right triangle. So this is called
a right triangle. Now, with the Pythagorean
theorem, if we know two sides of a right triangle we can
always figure out the third side. And before I show you how to
do that, let me give you one more piece of terminology. The longest side of a right
triangle is the side opposite the 90 degree angle-- or
opposite the right angle. So in this case it is
this side right here. This is the longest side. And the way to figure out where
that right triangle is, and kind of it opens into
that longest side. That longest side is
called the hypotenuse. And it's good to know, because
we'll keep referring to it. And just so we always are good
at identifying the hypotenuse, let me draw a couple of
more right triangles. So let's say I have a triangle
that looks like that. Let me draw it a
little bit nicer. So let's say I have a triangle
that looks like that. And I were to tell you
that this angle right here is 90 degrees. In this situation this is the
hypotenuse, because it is opposite the 90 degree angle. It is the longest side. Let me do one more, just
so that we're good at recognizing the hypotenuse. So let's say that that is my
triangle, and this is the 90 degree angle right there. And I think you know how
to do this already. You go right what
it opens into. That is the hypotenuse. That is the longest side. So once you have identified the
hypotenuse-- and let's say that that has length C. And now we're going to
learn what the Pythagorean theorem tells us. So let's say that C is equal to
the length of the hypotenuse. So let's call this
C-- that side is C. Let's call this side
right over here A. And let's call this
side over here B. So the Pythagorean theorem
tells us that A squared-- so the length of one of the
shorter sides squared-- plus the length of the other shorter
side squared is going to be equal to the length of
the hypotenuse squared. Now let's do that with an
actual problem, and you'll see that it's actually not so bad. So let's say that I have a
triangle that looks like this. Let me draw it. Let's say this is my triangle. It looks something like this. And let's say that they tell us
that this is the right angle. That this length right here--
let me do this in different colors-- this length right
here is 3, and that this length right here is 4. And they want us to figure
out that length right there. Now the first thing you want to
do, before you even apply the Pythagorean theorem, is to
make sure you have your hypotenuse straight. You make sure you know
what you're solving for. And in this circumstance we're
solving for the hypotenuse. And we know that because this
side over here, it is the side opposite the right angle. If we look at the Pythagorean
theorem, this is C. This is the longest side. So now we're ready to apply
the Pythagorean theorem. It tells us that 4 squared--
one of the shorter sides-- plus 3 squared-- the square of
another of the shorter sides-- is going to be equal to this
longer side squared-- the hypotenuse squared-- is going
to be equal to C squared. And then you just solve for C. So 4 squared is the same
thing as 4 times 4. That is 16. And 3 squared is the same
thing as 3 times 3. So that is 9. And that is going to be
equal to C squared. Now what is 16 plus 9? It's 25. So 25 is equal to C squared. And we could take the positive
square root of both sides. I guess, just if you look at
it mathematically, it could be negative 5 as well. But we're dealing with
distances, so we only care about the positive roots. So you take the principal
root of both sides and you get 5 is equal to C. Or, the length of the
longest side is equal to 5. Now, you can use the
Pythagorean theorem, if we give you two of the sides, to figure
out the third side no matter what the third side is. So let's do another
one right over here. Let's say that our
triangle looks like this. And that is our right angle. Let's say this side over here
has length 12, and let's say that this side over
here has length 6. And we want to figure out this
length right over there. Now, like I said, the first
thing you want to do is identify the hypotenuse. And that's going to be the side
opposite the right angle. We have the right angle here. You go opposite
the right angle. The longest side, the
hypotenuse, is right there. So if we think about the
Pythagorean theorem-- that A squared plus B squared is
equal to C squared-- 12 you could view as C. This is the hypotenuse. The C squared is the
hypotenuse squared. So you could say
12 is equal to C. And then we could say that
these sides, it doesn't matter whether you call one of
them A or one of them B. So let's just call
this side right here. Let's say A is equal to 6. And then we say B-- this
colored B-- is equal to question mark. And now we can apply the
Pythagorean theorem. A squared, which is 6 squared,
plus the unknown B squared is equal to the hypotenuse
squared-- is equal to C squared. Is equal to 12 squared. And now we can solve for B. And notice the difference here. Now we're not solving
for the hypotenuse. We're solving for one
of the shorter sides. In the last example we
solved for the hypotenuse. We solved for C. So that's why it's always
important to recognize that A squared plus B squared plus C
squared, C is the length of the hypotenuse. So let's just solve for B here. So we get 6 squared is 36,
plus B squared, is equal to 12 squared-- this
12 times 12-- is 144. Now we can subtract 36 from
both sides of this equation. Those cancel out. On the left-hand side we're
left with just a B squared is equal to-- now 144
minus 36 is what? 144 minus 30 is 114. And then you
subtract 6, is 108. So this is going to be 108. So that's what B squared is,
and now we want to take the principal root, or the
positive root, of both sides. And you get B is equal
to the square root, the principal root, of 108. Now let's see if we can
simplify this a little bit. The square root of 108. And what we could do is
we could take the prime factorization of 108
and see how we can simplify this radical. So 108 is the same thing as 2
times 54, which is the same thing as 2 times 27, which is
the same thing as 3 times 9. So we have the square root of
108 is the same thing as the square root of 2 times 2
times-- well actually, I'm not done. 9 can be factorized
into 3 times 3. So it's 2 times 2 times
3 times 3 times 3. And so, we have a couple of
perfect squares in here. Let me rewrite it a
little bit neater. And this is all an exercise in
simplifying radicals that you will bump into a lot while
doing the Pythagorean theorem, so it doesn't hurt to
do it right here. So this is the same thing as
the square root of 2 times 2 times 3 times 3 times the
square root of that last 3 right over there. And this is the same thing. And, you know, you wouldn't
have to do all of this on paper. You could do it in your head. What is this? 2 times 2 is 4. 4 times 9, this is 36. So this is the square root of
36 times the square root of 3. The principal root of 36 is 6. So this simplifies to
6 square roots of 3. So the length of B, you could
write it as the square root of 108, or you could say it's
equal to 6 times the square root of 3. This is 12, this is 6. And the square root of 3,
well this is going to be a 1 point something something. So it's going to be a
little bit larger than 6.