Hi, I’m Rob. Welcome to Math Antics!
In this lesson, we’re gonna learn about The Pythagorean Theorem
or Pythagoras’ Theorem as it’s sometimes called.
And you may be wondering, “What’s a Theorem?”
and “Who in the world is Pythagoras?”
Well, in math, a theorem is simply a statement that has been proven to be true
from other things that are either known or accepted to be true.
And Pythagoras… well, he was this really smart dude
who lived a long time ago in ancient Greece
and he proved the theorem. [Applause]
Well… historians aren’t completely sure it was actually Pythagoras that proved it.
It could have been one of his students or followers,
but he usually gets credit for it.
[applause & cheers]
Anyway, the main thing that you need to know is that
the Pythagorean Theorem describes an important geometric relationship
between the three sides of a right triangle.
We’re gonna learn what that relationship is in just a minute,
but first there's several things that you’ll need to know
before you can truly understand the Pythagorean Theorem or use it to solve problems.
First of all, to understand the Pythagorean Theorem you need to know about angles and triangles,
and you also need to know a little bit about exponents and square roots.
So if those topics are new to you, be sure to watch our videos about them first.
And second, even though the Pythagorean Theorem is about geometry,
you’ll need to know some basic algebra to actually use it.
Specifically, you’ll need to know about variables
and how to solve basic algebraic equations that involve exponents.
We cover a lot of those topics in the first five videos of our Algebra Basics series.
Okay, now that you’ve got all that background info covered,
let’s see what the Pythagorean Theorem actually says.
The theorem can be stated in several different ways,
but the one we like best goes like this:
For a right triangle with legs ‘a’ and ‘b’ and hypotenuse ‘c’,
‘a squared’ plus ‘b squared’ equals ‘c squared’.
As you can see from this definition, the Pythagorean Theorem doesn’t apply to ALL triangles.
It ONLY applies to RIGHT triangles.
As you know, right triangles always include one right angle
that’s usually marked with the square ‘right-angle-symbol’ to help you identify it.
And you need to know which angle is the right angle because
it helps you identify an important side of the triangle called the hypotenuse.
The hypotenuse is the longest side of a right triangle
and it’s always the side that’s ‘opposite’ of the right angle.
In other words, it’s the side that doesn't touch (or help form) the right angle itself.
In order to use the Pythagorean Theorem, you need to be able to identify the hypotenuse
because that’s what the variable ‘c’ stands for in theorem.
’c’ is the length of the hypotenuse side.
The other two sides of the triangle
(the ones that DO touch or form the right angle) are called its “legs”.
Our Pythagorean Theorem definition uses the variable names ‘a’ and ‘b’ to represent their lengths.
Oh… and it does’t matter which leg is called ‘a’ and which leg is called ‘b’
as long as you keep track of which is which after you make your initial choice.
Okay, now that we know the various parts of the Pythagorean Theorem,
let’s think about what the relationship or equation ('a squared' plus 'b squared' equals 'c squared') is really telling us.
It’s telling us that if we take the lengths of the two legs (sides ‘a’ and ‘b’) and ‘square’ them,
which means multiplying them by themselves,
(‘a squared’ is ‘a’ times ‘a’ and ‘b squared’ is ‘b’ times ‘b’)
…and then if we add those two ‘squared amounts’ together,
they will EQUAL the amount you’d get if you ‘square’ the hypotenuse side
(which would be ‘c squared’ or c times c)
That may sound a little confusing at first.
So let’s take a look at a special example of a right triangle
that will help the Pythagorean Theorem make a little more sense.
This right triangle is called a “3, 4, 5 triangle”
because its sides have relative lengths of 3, 4, and 5.
And by “relative lengths”, I mean that the units of length don’t really matter.
The sides could be expressed in ANY units (inches, meters, miles, whatever…)
So the triangle could be of any size as long as its lengths
would have the proportions 3, 4, and 5 relative to each other.
Starting with the side that’s 3 units long, which we’ll call “side a”
what do we get if we square that side?
Well in arithmetic, squaring 3 mean multiplying 3 times 3 which equals 9.
And the geometric equivalent of squaring something actually results in a square shape.
As you can see, this square contain 9 unit squares.
So this red area represents the value ‘a’ squared in the Pythagorean Theorem.
Next let’s look at the side that’s 4 units long, which we’ll call “side b”.
Squaring 4 means multiplying 4 times 4 which is 16.
Again, the geometric equivalent of that is a literal square
that is 4 units on each side and covers a total area of 16 units.
So this blue area represent ‘b squared' in the Pythagorean Theorem.
And finally, let’s deal with the hypotenuse, or “side c”, which is the longest side.
It’s 5 units long. Squaring 5 means multiplying 5 times 5 which is 25,
and the geometric equivalent is a 5 by 5 square that has an area of 25 units.
So this green area represents ‘c squared’ in the Pythagorean Theorem.
Now that you can see how the arithmetic parts of the Pythagorean Theorem
are related to the geometric parts of this right triangle,
let’s check to see if the Pythagorean Theorem is really true (at least in this special case).
On the arithmetic side, if you add up the amounts ‘a squared’ and ‘b squared’,
they really DO equal ‘c squared’ because 9 + 16 = 25.
And… with a little rearranging of our unit squares,
you can see that the area of the squares formed by the two legs
really does equal the area of the square formed by the hypotenuse.
Wow, those ancient greek dudes really were smart!
Okay, but I know what some of you are thinking…
“That’s cool ’n all, but… why should I even care about the Pythagorean Theorem?
What’s it good for?”
Well, as always, that’s a good question.
And the answer is, like many things in math,
the Pythagorean Theorem is a useful tool
that can help you use what you DO know to figure out what you DON’T know.
Specifically, if you have a right triangle but you only know how long two of its sides are,
the Pythagorean Theorem tells you how to figure out the length of the third unknown side.
For example, imagine that you have a right triangle
that’s 2 cm long on this side and 3 cm long on this side,
but we don’t know how long the hypotenuse is.
No problemo!
The Pythagorean Theorem tells us the relationship between all three sides of ANY right triangle,
so we can figure it out.
We know that 'a squared' plus 'b squared' equals 'c squared'
so let’s plug what we DO know into that equation
and then solve it for what we DON’T know.
Again, it doesn’t matter which of the two legs is called ‘a’ or ‘b’
so let’s just label them like this
and then substitute ‘2’ for ‘a’ and ‘3’ for ‘b’ in the Pythagorean Theorem equation.
That gives us an algebraic equation that has just one unknown, ‘c’.
If we solve this equation for ‘c’… in other words,
if we re-arrange the equation so that ‘c’ is all by itself on one side of the equal sign,
then we’ll know exactly what ‘c’ is.
We’ll know the length of that side of the triangle.
First, we need to simplify the left side of the equation since it contains the known numbers.
And according to the order of operations, we need to simplify the exponents first.
‘2 squared’ is 4 and ‘3 squared’ is 9.
Then, we add those results (4 + 9 = 13)
and we have the equation 13 = ‘c squared’
which is the same as ‘c squared’ = 13.
Then, to get ‘c’ all by itself,
we need to do the inverse of what’s being done to it.
Since it’s being squared, the inverse operation is the square root,
so we need to take the square root of both sides.
Taking the square root of 'c squared' just gives us ‘c’ which is what we want on this side of the equation,
but it gives us a little problem on the other side
because it’s not easy to figure out what the square root of 13 is.
It’s not a perfect square so it’s going to be a decimal,
and probably an irrational number.
But that’s okay because it’s fine to just leave our answer as the ‘square root of 13’
Sure, you could use a calculator to get the decimal value if you really need one,
but in math it’s very common to just leave square roots alone unless they're easy to simplify.
So the sides of this right triangle are, 2 cm, 3 cm and the ‘square root of 13’ cm.
Let’s try another example.
For this right triangle, we know the length of the hypotenuse (6 m),
and one of the legs (4 m),
but the length of the other leg is unknown.
So let’s use the Pythagorean Theorem to find that unknown length.
As usual, we call the hypotenuse “side c”.
And let’s call the leg we know “side a” and the leg we don’t know “side b”.
Then we can substitute the known values into the Pythagorean Theorem
and solve for the unknown value.
Replacing the ‘c’ with 6 and the ‘a’ with 4 gives us the equation,
‘4 squared’ plus ‘b squared’ equals ‘6 squared’,
which we need to simplify and solve for ‘b’.
First let’s simplify the exponents.
‘4 squared’ is 16, and ‘6 squared’ is 36.
Now we need to isolate the ‘b squared’
and we can do that by subtracting 16 from both sides of the equation.
On this side, the (+16) and the (-16) leave us with just ‘b squared’.
And on the other side we have 36 minus 16 which is 20.
We can now solve this simplified equation for ‘b’
by taking the square root of both sides which gives us
‘b’ equals the ‘square root of 20’.
Again, it’s fine to leave your answer as a square root like this.
And some of you may know that the ‘square root of 20’ can be simplified
to ‘2 times the square root of 5’.
We’re not going to worry about simplifying roots in this video,
but if you know how to do it, awesome!
If you don’t know, just leave the answer as the ‘square root of 20’ meters.
Here’s another interesting one.
What if you have a ‘unit square’ that's cut in half along a diagonal.
Each side of the square is 1 unit long,
but how far is it from one corner of the square to the other along the diagonal?
Well, since the diagonal divides the square into two right triangles,
we can use the Pythagorean Theorem to tell us that unknown distance.
We label the legs of the right triangle ‘a’ and ‘b’ and the hypotenuse ‘c’.
And since we know that ‘a’ and ‘b’ are both 1, we can plug those values
into the Pythagorean Theorem equation which gives us us
‘1 squared’ plus ‘1 squared’ equals ‘c squared’.
Now we solve for ‘c’.
‘1 squared’ is just 1
so this left side of this equation simplifies to 1 plus 1 which is just 2.
That means ‘c squared’ equals 2
and if we take the square root of both sides,
we get ‘c’ equals the ‘square root of 2’.
So that’s how far it is across the diagonal of the unit square.
Okay, so that’s how you use the Pythagorean Theorem
to find the length of an unknown side of a right triangle, which is its most common use.
But there’s another way that you can use the Pythagorean Theorem that I want to mention.
You can also use the Pythagorean Theorem to TEST a triangle
to see if it truly is a RIGHT triangle
Ya know… in case you’re not already sure.
For example, what if someone shows you this triangle and asks,
“Is this a right triangle?”.
Well, it looks a lot like a right triangle
but it does’t have a right angle symbol,
and it would be kinda hard to tell if this angle is exactly 90 degrees just by looking at it.
Maybe it’s really close to 90, like 89.5 degrees.
No worries, the Pythagorean Theorem can tell us for sure,
if we know the lengths of all three sides of the triangle.
If we know those lengths (a, b and c),
then we can just plug them into the Pythagorean Theorem equation to see if it holds true.
In this particular case,
since the two shorter sides are each 3 cm and the longest side is 4 cm,
we can plug those values in for ‘a’, ‘b’ and ‘c’ and simplify to see what we get.
‘3 squared’ is 9, so on this side of the equation we get 9 plus 9 which is 18.
And on the other side we have ‘4 squared’, which is 16.
Uh oh… that doesn’t look right!
Our equation simplified to 18 equals 16,
which is definitely NOT a true statement.
That means that the three sides of this triangle do not work in the Pythagorean Theorem…
they don’t fit the relationship 'a squared' plus 'b squared' equals 'c squared'
And since the Pythagorean Theorem tells us that ALL right triangle fit that relationship,
this triangle must not be a RIGHT triangle.
Alright…. so now you know what the Pythagorean Theorem is and you know how to use it.
You can use it to find a missing side of any right triangle,
and you can also use it to test a triangle to see if it qualifies as a right triangle.
But as you can see,
it takes a lot of other math skills to be able to use the Pythagorean Theorem effectively,
so you may need to brush up on some of those skills before you’re ready to try using it on your own.
And remember, you can’t get good at math just by watching videos about it.
You actually need to practice solving real math problems.
As always, thanks for watching Math Antics and I’ll see ya next time.
Learn more at www.mathantics.com