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The Pythagorean theorem is one of the most useful tools you'll ever learn mostly because

you just use it so often.

You can't get through algebra, geometry, trig, or even calculus without it.

And in this video we're going to be talking about everything related to the Pythagorean

theorem, including what it is, why it actually works, and how you can use it in real life.

So what is the Pythagorean theorem?

well first of all it's a theorem that only works for right triangles.

It doesn't work for obtuse triangles or acute triangles and we'll talk about that more in

a bit.

It can only be applied when you have a right triangle.

Let's remember that a right triangle is a triangle with one 90 degree angle which we

indicate here with this little box in one angle of the triangle.

So this little box means that it's a 90 degree angle.

And when a triangle has exactly one angle that measures exactly 90 degrees, so not 89,

not 91, but 90, that is a right triangle.

And the Pythagorean theorem only works for right triangles.

So we have one angle that's 90 degrees.

It doesn't matter the value of the other two angles.

They could be anything as long as one angle is 90 degrees it's a right triangle and we

can apply the Pythagorean theorem.

So under those circumstances what does the Pythagorean theorem state?

Well it's a relationship between the length of each side of this right triangle.

So if we were to call the length of this side a, if we called the length of this side b,

and if we called the length of the hypotenuse which remember is always the longest side

and it's always the side that's opposite your 90-degree angle, so I find my 90 degree angle

and I go straight across to the other side of the triangle, that is the hypotenuse, it's

the longest side, so if we call the hypotenuse c, then the Pythagorean theorem says that

if I take the length of side a and I square it, so I get a^2 and then I add to that b^2,

so I take the length of side b and I square that and I get b^2, when I add these two things

together that will always be equal to c^2, the length of the hypotenuse squared.

And the Pythagorean theorem and right triangles are inextricably tied together, they're always

a pair because the Pythagorean theorem works for every single right triangle you will ever

find.

It doesn't matter what the length of side a is or the length of side b or the length

of side c.

If it's a right triangle then the Pythagorean theorem a^2+b^2=c^2 will always be true.

And normally when we talk about the Pythagorean theorem we like to think of it as starting

with the triangle and then the Pythagorean theorem is a result.

In other words sort of this logic that if we have a right triangle, if we have a triangle

with a 90-degree angle, then we know that the Pythagorean theorem is going to be true

for that triangle.

But we can sort of think of it the opposite way too and this is the converse of the Pythagorean

theorem.

The logic is basically the opposite which says that...

Pretend I don't have this triangle at all right now I just know that I have a triangle

and I'm calling the sides a, b, and c.

Well if I know for that triangle that a^2+b^2=c^2 then I automatically know that I have a right

triangle, that the triangle is a right triangle.

So you can sort of start with this theorem and conclude that you have a right triangle

or the opposite way, know that you have a right triangle up front and conclude that

this theorem has to be true.

So the Pythagorean theorem is that given a right triangle I know that a^2+b^2=c^2.

The converse of the Pythagorean theorem is if I know that a^2+b^2=c^2 then I know that

I have a right triangle.

Now remember before we talked about acute and obtuse triangles.

This is a right triangle but at the most broad level there are three kinds of triangles,

a right triangle with a 90 degree angle, an acute triangle, and an obtuse triangle.

In an acute triangle the largest angle is less than 90 degrees.

That might look something like this.

All three of these angles in this triangle are less than 90 degrees so this is an acute

triangle.

Or you can have an obtuse triangle where the largest angle is greater than 90 degrees and

that might look something like this.

The largest angle in this triangle is this angle here and it's larger than 90 degrees.

So you have an acute triangle, a right triangle, and an obtuse triangle.

And the Pythagorean theorem is sort of a midpoint.

It says that in a right triangle a^2+b^2 is exactly equal to c^2.

Well when we have an acute triangle we know that a^2+b^2 is going to be greater than c^2.

And when we have an obtuse triangle we know that a^2+b^2 is less than c^2.

So you can almost think of this like the Goldilocks story where everything is either too small,

too big, or just right.

In an acute triangle a^2+b^2 is bigger than c^2.

In an obtuse triangle a^2+b^2 is less than c^2.

But just right in the middle of those is the right triangle where a^2+b^2 is exactly equal

to c^2 every time.

And that's why Pythagorean theorem is only for right triangles because if your largest

angle is less than 90 degrees and you have an acute triangle then this inequality gets

skewed one way.

If your largest angle is greater than 90 degrees and you have an obtuse triangle the inequality

gets skewed the other way.

Pythagorean theorem is only for right triangles because it's only when you have that 90-degree

angle that this equation happens to be exactly spot-on and a^2+b^2 will in fact equal c^2.

So how does the Pythagorean theorem actually work?

Well let's go ahead and walk through a proof together.

We'll actually do the proof that Pythagoras himself used when he proved this theorem.

Pythagoras was a Greek mathematician and philosopher who lived between about 570 and 500 BC and

he's actually thought to be the first person to write down the proof of the theorem, but

people had knowledge of it and we're using it long before Pythagoras lived including

the Babylonians as early as about 1900 BC or so.

So why is the Pythagorean theorem true?

Well let's try to prove it right now.

If I have two squares and these are identical because I just copied one and pasted it so

these have exactly the same area and the same dimensions, they're both the same height and

the same width.

So if I draw a vertical line in this first square and I can actually draw this anywhere

let's say I do it here and then I make a perfectly horizontal line maybe about here, such that

this little square is a perfect square and this larger square is a perfect square.

And then I want to go ahead and bisect these two rectangles here so that I split them both

into two triangles.

Now at this point I have these four triangles and I want to go ahead and look at one of

them.

So if I look at this triangle right here I can see that it's a right triangle because

this is a perfectly horizontal line and this is a perfectly vertical line which means that

this is a right triangle with my 90 degree angle right here.

So I've got a right triangle.

I can call the length of this side of it a and the length of this side b.

That means that this perfectly symmetrical triangle that's sitting on top of it, it's

the exact same dimensions it's just flipped over, this has dimensions here a and b as

well.

Now remember this little square is a perfect square so I know this is a by a, I know the

length of all of its sides are a.

And this is also a perfect square so I know that the length of all of the sides of this

square are b.

Which means that when I look at these other two triangles here I can see that the length

of this side is b, I already know that, and the length of this little side then is a based

on the length of or the width of the square up here.

So this is also a and this triangle right here on the right, this is a times b.

So notice here that by Pythagorean theorem I have four triangles, the lengths of the

legs are a and b and the length of the hypotenuse I can call c.

So this is c and c obviously and then c and c here.

Now how do I prove that a^2+b^2=c^2 for one of these triangles?

Well if I take the size of this triangle, one of these triangles, and I translate it

directly over into this other square here then my triangle looks about like this and

I can actually draw four of these around the edges of the triangle that have the same dimensions

as the one in the triangle on the left.

So all four of these triangles around the outside I can now say have dimensions a and

b here, a and b here, a and b here, and a and b.

I also know that the length of each hypotenuse of the triangle is c which I get from this

other square so I can say c, c, c, and c.

Now I just need to add up the area of each full square.

So if I look at the square over here on the left I know I have this little square, a times

a, so the area of this small square is a^2, so I get a^2.

The area of this large square is b times b or b^2.

The area of each triangle, remember the formula for the area of a triangle is one half times

the length of the base times the length of the height so that would be one half times

b times a, or one half times a times b.

And I have four triangles so I want to say I have four triangles and the area of each

one is (1/2)ab.

Now over here on the right I have the area of the square in the center which is c times

c or c^2 so I get c^2.

And then I know that the area of each triangle, first of all I have four of them, so I want

to say total area is 4... and then the area of each triangle is 1/2 times a times b.

And I already know the area of these two squares is equal.

So if I were to set these equal to one another, I have a 4 times 1/2 times ab on each side,

so I could subtract that out from both sides and these values would cancel away completely.

And then all I'm left with on the left side is a^2+b^2 and all I'm left with on the right

side is c^2 so I know by this proof that a^2+b^2 must be equal to c^2.

And that's how Pythagoras himself proved that the Pythagorean theorem was true for right

triangles.

So now the last question that remains is how do we actually use Pythagorean theorem and

how can it be used in real life?

Well its most common and simple application is just finding the distance between two things

when those things are on a diagonal.

For example let's pretend that I need to do some roofing work on my house.

I need to get up on the roof and make some fixes to the roof and I need to know how long

of a ladder I need to get in order to reach the roof.

That's a simple problem that I can actually solve with Pythagorean theorem.

So let's pretend that this is my house with my roof on top of it and I need to be able

to reach to this point right here from my ladder on the ground.

So if I say for example that this has to be what my ladder looks like, how do I figure

out how long the ladder needs to be in order for me to reach the roof?

Well it would be difficult to calculate the length of this diagonal or to measure it through

space if I try to take a measuring tape and measure this distance that would be very difficult.

What would be much easier is if I could recognize that this is just the hypotenuse of triangle.

So instead I construct a triangle and I go here parallel to my house and then along the

ground like this.

And I know right away that since I have this distance level along the ground and I have

this distance which is perfectly vertical that this is a right triangle because these

two lines are perpendicular.

This is perfectly vertical, perfectly horizontal, which means this is a 90 degree angle and

this is a right triangle and that means that because it's a right triangle I can use Pythagorean

theorem.

Let's say that I already know that my house is 12 feet tall which is maybe a safe assumption

because I know how far it is from the floor to the ceiling in my house so let's say I

know that this is 12 feet here.

And let's say I want to place my ladder, the base of it, 5 feet away from this point here.

I could easily measure that with a measuring tape along the ground and so I could say that

this is 5 feet here.

Now I can easily use the Pythagorean theorem to find the length that the latter needs to

be to reach this point.

So again this is my hypotenuse, it's the longest side across from the 90 degree angle.

So then I can use Pythagorean theorem and say that I want to square this side, so I

want to say 5^2.

I know that by Pythagorean theorem when I add that to the square of this side, so 12^2,

that that has to be equal to the length of the hypotenuse squared.

That has to be equal to c^2.

And now here's how I solve this for c.

I take 5^2 is 25, so 25.

I take 12^2 is 144.

And then I have c^2.

25 plus 144 is 169.

Now in order to solve for c, I need to take the square root of both sides.

So I want to take the square root of 169 and the square root of c^2.

Now the square root of c^2 is c.

And the square root of 169 is either positive 13 or negative 13.

But remember with Pythagorean theorem we're always dealing with right triangles and it's

impossible for the length of any side of a triangle to be negative and for that triangle

to still exist.

So we can always ignore that negative value and we can say that c is equal to positive

13.

Therefore I know then that the length of my ladder has to be 13 feet in order for me to

safely reach from the ground to the corner of the roof if I place the base of the ladder

5 feet away from the house.

The other great thing about Pythagorean theorem is that you can use it to solve for any side

of the triangle.

So in this real-world problem we did not know the value of the hypotenuse but we knew the

value of the other two legs of this triangle.

But let's say that instead we already knew that the ladder was 13 feet long and we already

knew that the height of the house was 12 feet and we wanted to know how far away from the

house we could place the base of the ladder.

In other words how far can this distance be?

Well instead of this equation here this would be our unknown value so we would call that

a and we would say a^2.

We already know 12 so we would say 12^2.

And then we already know the length of the ladder is 13.

And then we could solve this for a by saying a^2 plus 144 is equal to 169.

Then we would subtract 144 from both sides and we would get a^2 is equal to 25.

And then we would take the square root of both sides order to solve for a so the square

root of a^2 is a and the square root of 25 is positive or negative 5 but again remember

that because we're dealing with a triangle we have to have a positive distance so we

can forget about the negative value and just say positive 5.

So it doesn't matter which two sides of the triangle you have.

You could have 5 and 12, you could have 12 and 13, or you could have 13 and 5.

If you just plug them in to the correct places in the Pythagorean theorem formula you'll

always be able to solve for the length of the third side.

We'll talk about one more thing as it relates to the Pythagorean theorem and that is what's

called Pythagorean triples.

Pythagorean triples are sets of three integers that satisfy the Pythagorean theorem.

So in a lot of cases you'll be given two values, let's say for example the values for a and

b, and you'll need to solve for the value of c.

But when you plug those things into the Pythagorean theorem and you solve for c you'll end up

with either a decimal or fraction value for c, it won't be a perfect integer, it won't

be a whole number.

Same thing goes if you're trying to solve for one of the other side lengths.

There are an infinite number of values that satisfy the Pythagorean theorem for a, b,

and c, but not every set is going to be three integers where a and b and c are all positive

whole numbers.

A Pythagorean triple on the other hand will be three integers, three positive whole numbers

that satisfy the Pythagorean theorem.

The simplest and most famous one if we have values here of a, b, and c, is the set 3,

4, 5.

Now there's a couple of things to keep in mind about Pythagorean triples.

The first is that a always needs to be what we call the opposite side.

a is always first of all the shortest side in the right triangle.

So c is always the hypotenuse, the longest side and the side that's across from the 90

degree angle.

And then you have the other two legs of the triangle.

And the shortest leg, the shortest side, is always going to need to be a.

So if we keep that rule and we say a is always the shortest side then when a is 3, the shortest

side is 3, then b the other leg is 4 and c the hypotenuse is 5.

And we can see that if we plug this into the Pythagorean theorem that this set of values

satisfies the Pythagorean theorem because we get 3^2 plus 4^2 equals 5^2.

And when we simplify this we get 3^2 is 9, 4^2 is 16, and 5^2 is 25.

9 plus 16 is 25 so we get 25 equals 25 and we can see that the set 3, 4, 5, satisfies

the Pythagorean theorem because this equation is true.

Now the cool thing about Pythagorean triples is that you can pick any odd number for a.

So 3, 5, 7, 9, 11, etc., any odd number, and you will have a Pythagorean triple.

Now it's important to say that you can pick a value for b or c, for example if I have

a value of 5 for a, the shortest side, I can then pick any value for b that's greater than

5 since a is the shortest side b has to be larger than that, I could pick a value for

b, and c would not necessarily be a positive whole number.

For example if I picked b equals 6 I would not get a positive whole number for c so that

would not be a Pythagorean triple.

It's just that a Pythagorean triple is possible, there is one possible Pythagorean triple,

for every odd value of a.

Here's how you find the specific Pythagorean triple for every odd number a.

b is always going to be a^2 minus 1 all divided by 2.

And c the hypotenuse is always going to be b plus 1.

So that's a lot right there so what do we actually mean by that?

Well here's what it means, let's dissect one of these.

Let's take 5.

Remember we can pick any positive odd whole number for a so 3, 5, 7, 9, 11, 13, 15, any

odd number for a the shortest side and we know we're going to come up with a Pythagorean

triple.

So let's say we pick 5 and now we want to know what b and c are that match for that

Pythagorean triple.

Well we take the value of a 5 and we plug it into this a^2 minus 1 over 1.

So what we get there is 5^2 minus 1 over 2 which is equal to 25 minus 1 in the numerator,

well 25 minus 1 is 24.

And then 24 divided by 2 is 12, so 12.

And then to get the value for c all I need to do is take b and add 1 to it because the

formula for c here is b plus 1.

I know b is 12 so 12 plus 1 is 13.

So my Pythagorean triple for a, b, and c is 5, 12, 13.

Which should look familiar because remember this was the Pythagorean triple we used in

the last problem where we were talking about the ladder leaning up against the house.

So because these are three positive whole numbers, they're integers, they form a Pythagorean

triple.

And I can do the same thing with any other positive odd whole number for a so 7, 9, 11,

etc.

I could create a Pythagorean triple using these formulas.

I hope that video helped you and if it did, hit that like button, make sure to subscribe,

and I'll see you in the next video.

For more infomation >> Pythagorean theorem - What is it? - Duration: 24:45.

-------------------------------------------

Eugene Robinson Writes That President Trump Is 'Out of Control' | Morning Joe | MSNBC - Duration: 5:32.

>>> WELCOME BACK TO "MORNING JOE."

JOINING US FOR THE MUST-READ OP EDS, ASSOCIATE EDITOR OF "THE

WASHINGTON POST" AND MSNBC POLITICAL ANALYST EUGENE

ROBINSON. YOU CAME ON A GOOD DAY.

WE'RE READING YOUR OP ED. >> HOW ABOUT IT, WILLIE?

>> YOU WRITE THAT TRUMP IS OUT OF CONTROL.

GENE WRITES THE STATEMENTS PRESIDENT TRUMP ISSUED ON

TWITTER IN RECENT DAYS LEAD TO A CHILLING CONCLUSION.

THE MAN IS OUT OF CONTROL. I KNOW THAT'S A RADICAL THING TO

SAY ABOUT THE ELECTED LEADER OF THE UNITED STATES, THE MOST

POWERFUL INDIVIDUAL IN THE WORLD.

I KNOW HIS UNORTHODOX USE OF SOCIAL MEDIA IS THOUGHT BY SOME,

INCLUDING THE PRESIDENT HIMSELF, TO BE BRILLIANT.

BUT I DON'T SEE POLITICAL GENIUS IN THE INVECTIVE COMING FROM

TRUMP THESE DAYS. I SEE AN ANGRY MAN LASHING OUT

AT ENEMIES REAL AND IMAGINED. WE ALREADY KNEW HE HAD A NARROW

MIND AND A SMALL HEART. NOW WE MUST WONDER ABOUT HIS

EMOTIONAL STABILITY, HIS GRASP OF REALITY OR BOTH.

GENE, THAT'S ALONG THE LINES OF SOME OF THE THINGS SAID AROUND

THE TABLE, HE'S NOT JUST ACTING OUT BUT AGAINST HIS OWN

SELF-INTEREST WITH MANY OF THESE TWEETS.

>> WHAT ELSE CAN YOU SAY, WILLIE.

AT FIRST, ONE POINT THAT REALLY HAS TO BE MADE IS THESE TWITTER

TWEET STORMS. WE CALL THEM TWEETS AND THAT

MINIMIZES THEM. THESE ARE STATEMENTS FROM THE

PRESIDENT OF THE UNITED STATES. THESE ARE STATEMENTS WRITTEN BY

THE HAND, BY THE TINY THUMBS OF THE PRESIDENT OF THE UNITED

STATES AND THEY'RE GOING TO HAVE TO BE TAKEN INTO ACCOUNT, I

BELIEVE, BY THE SUPREME COURT, FOR EXAMPLE, WHEN IT DECIDES ON

THE TRAVEL BAN. YOU COULD ARGUE THAT WHAT HE

SAID DURING THE CAMPAIGN, MAYBE YOU CAN FIND SOME WAY TO

OVERLOOK IT. YOU CAN'T OVERLOOK WHAT HE SAYS

AS A SITTING PRESIDENT OF THE UNITED STATES.

IT'S ABOUT HIS OWN EXECUTIVE ORDER.

I THINK THAT ISSUE PROBABLY IS OFF THE TABLE FOR HIM.

HE KEEPS DAMAGING HIMSELF, AGAIN, WHAT HE SAID ABOUT

LONDON, IT'S JUST OUTRAGEOUS AND OFFENSIVE TO ME AS AN AMERICAN

AS YOU LOOK AT WHAT HAPPENED IN LONDON AND HOW IT'S BEEN HANDLED

BY MAYOR KHAN OVER THERE. IT'S AS IF HE'S NOT IN CONTROL.

THAT'S WHY THE HEADLINE, HE'S OUT OF CONTROL.

>> NOT IN CONTROL, OUT OF CONTROL.

DONNY DEUTSCH, THERE'S A LINE HERE THAT GENE WROTE.

A MAN DANGEROUSLY OVERWHELMED. I THINK THAT'S OBVIOUS.

HE IS OVERWHELMED AND HE IS SO OVERWHELMED.

HE NOW UNDERSTANDS THAT HE UNDERESTIMATED HOW DIFFICULT

THIS WAS. ALL OF THEM, EVERYBODY AROUND

DONALD TRUMP UNDERESTIMATED JUST HOW DIFFICULT THIS JOB IS GOING

TO BE, AND NOW HE FINDS HIMSELF IN THE MIDDLE OF A STORM HAVING

TO GET LAWYERS, POSSIBLY FACING CRIMINAL SANCTIONS, AND HE IS

DANGEROUSLY OVERWHELMED. I GUESS WHAT IS MOST CONCERNING

IS THAT WHEN IN THIS DANGER, HE'S REACHING OUT AND GETTING

SUPPORT FROM STEVE BANNON, THE VERY GUY THAT HAS CAUSED MOST OF

THESE PROBLEMS, THE VERY MAN WHO HAS ISOLATED HIM NOT ONLY IN THE

WHITE HOUSE, BUT IN WASHINGTON, D.C. AND ACROSS THE WORLD.

>> YOU'RE TALKING ABOUT KING BANNON -- IS IT KING BANNON OR

PRESIDENT BANNON? I ALWAYS GET CONFUSED.

>> IT'S LITERALLY PRESIDENT BANNON BECAUSE HE'S BEEN SOLD A

BILL OF GOODS BY THIS HUXSTER SAYING, HEY, THIS IS WHAT YOU

NEED TO DO. PRESIDENT BANNON IS NOW SITTING

AT 36% IN THE POLLS, 37%. IT CAN'T GET MUCH WORSE.

THESE ARE HISTORICALLY LOW POLL NUMBERS FOR SOMEBODY AT THIS

STAGE OF THE PRESIDENCY. >> HERE IS MY QUESTION, AND YOU

TOUCHED ON THIS EARLIER. CLEARLY IN ALL DIFFERENT SHAPES

AND FORMS WE'RE SAYING THIS GUY IS CERTAINLY NOT WELL, NOT

MENTALLY STABLE. SO THERE ARE CERTAIN PEOPLE OF

GRAVITAS AROUND HIM, McMASTER, MATTIS, GARY KOHN, DINA POWELL.

I WONDER IF AT ANY POINT IN TIME AS PATRIOTS SOME OF THESE PEOPLE

RAISE THEIR HAND AND SAY SOMETHING IS NOT RIGHT HERE.

I KNOW ALL THESE PEOPLE ARE SOLID PEOPLE WITH GRAVITAS, WITH

RESUMES, WITH FAMILIES, AND OBVIOUSLY ARE WELL INTENTIONED.

YOU JUST WONDER DO ANY OF THOSE PEOPLE AT ANY POINT IN TIME FOR

THE GOOD OF THE COUNTRY, BECAUSE THEY CAN'T RAISE THEIR HAND TO

THE PRESIDENT, BUT RAISE THEIR HAND TO THE REST OF US THAT

SOMETHING IS TROUBLING HERE. >> DONNY, WE HAVE AN EVENT

COMING UP ON THURSDAY. WE HAVE JAMES COMEY'S TESTIMONY.

AND I WAS WONDERING YESTERDAY WHETHER, IN ADDITION TO WHATEVER

HE SAYS ABOUT HIS CONVERSATIONS WITH THE PRESIDENT ABOUT THE

RUSSIA THING, I ALSO WONDER IF HE WILL, FRANKLY, ADDRESS HIS

IMPRESSIONS OF THE PRESIDENT. THIS IS SOMEONE WHO WHO HAS SOM

GRAVITAS AND EXPERIENCE HERE. PEOPLE LIKE HIM AND DON'T LIKE

HIM, BUT HE'S A SERIOUS GUY.