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.

Thank you thank you thank you thank you!!!!!

People always made this sound so complicated, but its so simple. Either way someone explaining it as perfectly as you did helps a lot, thanks!

How can you use the Pythagorean theorem on a triangle that has two equal unknown sides

Ooooooo I get it

Did anyone notice he spelt angle wrong and spelt it like an angel 😇💀😂

In all pythagorean theorem numbers is

4! 3! 25! 5!

Make somethin different

Thanks I learned alot

I'm learning this in 8th grade :p

thx! i learned it! also i loved the colors :))

I'm gonna be honest I used to cry when i thought of math and the fact that I didn't understand stuff but now i don't I just finish H.W . like nun.

Interesting topic

I'm a 6th grader that's been doing 7th and 8th grade math in one school year.

I don't understand anything so ima prolly retake 8th grade math next year like a dummmyy

i was stressing over this for hours while I could just open yt oof

Ok this is GOOD.

my teacher just handed us this lesson and said take notes. for sure we have a quiz tomorrow and how can I take a quiz on something I don’t know? the only solution (pun intended) khan academy

Sir nice vedio

Pythagorean Theorem is hard

10.4 is the length of that side (B)

This video really helped me!! Thank you so much!!! They have practices too, it's in the description if you wanna try it.

I swear I hate school😩😩 I just want a job😂😂😂😂

It's so weird to see math in a different language, you guys even switch the numbers lol

8:09 to 8:10 you almost said sex

this makes math (dare I say it ) easy

well, i have SSAT, so i guess this is helpful. it's such a pain when i can be watching some random 5-minute crafts videos :,)

2019 hands off!

Thank you I needed to remind myself how to do this @ 2:21 A.M.

#YouCanLearnAnything

#VideoGames

#Top10Gaming

This guy like to hear himself talk

Thank you so much, I am in online school and my math grade is a C (in my family that is a failing grade), and this will really help me!

Why is this thought in older years. It seems straight forward. I'm in 7th grade/ year 8 and algebraic expressions and indices seem harder. 🤦🤷

Too easy.

Why was the last 3 excluded ?

Good teacher. Nice voice

What year level is this I’m in year 8 and need to study this

This is so dumb 🧟♀️🖕

Give me your talent. I would love this for math! Like this comment if you agree!

Oh yeah and sees🙂📓📕📔📒📗📚📚📖📘📙📝🔎1000

Isn't the theorem a form of proofing pie?

Nice video

Have my finals for math tmr 😰

Its pretty ez thx

Never underestimate the power of khan academy. I have a final exam tomorrow and I was crying earlier cuz I didn't remember or understand how to do all this stuff. now I've watched these videos I'm a pro. literally god bless khan academy you saved my sanity

I’m here because I heard Daniel Habif mention it in one of his videos. Is there something spiritual to it????

I was absent during this and I have my final today, thx, also your handwriting is really good

when you teach yourself the entire course the night before the exam lol. wish me luck, kids

Him: 16 + 9 = 25

Me: YES THATS SO EASY

Him: We get the roots

Me:

QuitsMy head is dead 💀 now

I've got a courage to master math, with khan.

Thank you for your nice math videos and I encourage you being a math teacher of

math

in South Korea's American math

academy and you would be a [email protected]!

The best thing is that these videos have a black background which helps in watching in the dark

It's school break but I'm still studying. ;-;

Somehow I'm learning more effectively because of these videos

8:59 this guy is a legend, me getting square root uses calculator

Don’t forget translate to arabic please

That was fun. Thank you!

Math is a drama queen, it can’t seriously have that many problems!

This is why I homeschool

who else is studying for the SHSAT?

Thank you so much! This has been impossible for the longest time, and now I feel so confident ☺️

Teacher: Triangle class, studie

Students: huh??

Edit: I’m in 5th grade so I don’t know what the heck is going on.

So my question is: Why is it called a therum? Numbers are factual. Aren't they??

Thanks Sal! You really helped

PIE TAH GO REE YAN TYO REMc=5

Why do you get 5?

can someone explain 10:01, why is he able to take the last three off? does he have to for the equation to work? please help.

Angel???

Mathematics is so funnnn!!! U make it seem so easy! Respect sal!!!!

The Pythagorean Theorem, woke me? When I went to technical school! Electronics. It Works, proves, and ___________, Fill in the blank.

Terrance Howard sent me hear SEP ”19”

Thank you! I was really struggling with a worksheet in math but now I completely get it.

Tysm, I was trying some problems involving this using just my math textbook and i got it super wrong and this helped explain it

thank you for helping me with my math stuff fellow human

I remember watching this 7 yers ago, and now I’m watching it again taking note on how to teach PYTH. THEO. to tutor middle school kids tomorrow. Idk how much I owe you from learning everything from you, ASTRONOMY, PHYSICS, CHEMISTRY, MATH, DUDE THEY GOT EVEN MUSIC lessons TOOO!!