[ Music ]
>> Hi, guys. Lindsay, here. And I want to take a few minutes today to talk about Pythagorean
theorem. And maybe you've watched our other video where we solved for the hypotenuse,
if not, you should watch it,
I think it's good. But what we're going to do today is we're actually going to take a
look at what if we're not solving for the hypotenuse?
What if we're looking for one of the other two sides?
And just to recap, on the hypotenuse, I'm going to call it H.
So, I don't have to write the word hypotenuse.
And the hypotenuse is always the side directly across from the right angle.
If you see what I've written up there, it says Pythagorean theorem can only be used
for right triangles.
A right angle, this little square guy, is a 90-degree angle.
If your triangle doesn't have a right angle in it, it doesn't have a hypotenuse.
Because you can't go across from a 90-degree angle that's not there.
So, this only works for a triangle with a right angle in it because it has a hypotenuse.
So again, here's Pythagorean theorem.
It's written lots of different ways.
The reason I write it this way is because H squared equals A squared plus B squared.
The H stands for hypotenuse. And it matters that the hypotenuse is by itself on one side
of the equal sign and then the other two sides are on the other side of the equal sign together.
So, I'm going to call this bottom guy A and I'll call this B. Although those are interchangeable.
But the hypotenuse isn't.
So, let's take a look at if we actually have some information on this triangle.
If my A value is, I'm just going to make these up here.
Let's say that that's 15 metres. Now let's say that in this situation, we know what the
hypotenuse is, let's say that the hypotenuse is 22 metres. And the question that I'm asking
then is find B. Find the side that you don't know.
So, let's take a look at Pythagorean theorem.
I'm going to write it down here again.
So, H squared equals A squared plus B squared.
And then I'm going to plug in the values that I have.
I have H. So, the biggest mistake I see when I see students doing questions like this is
they take the two sides that they know and they plug them in for A and B and they leave
this guy blank.
You can't do that. If you have the hypotenuse, that side across from the right angle, it's
got to go on one side of the equal sign by itself.
Which is what makes this question different from a question where we're finding the hypotenuse.
So, hypotenuse equals 22 metres. So where my H is I'm going to write 22.
With my squared above it. I decided that my A value is 15 metres.
So, 15 squared. And then my B value, I don't know it.
So, I'm just going to leave it as a B. So, I've plugged my values in and again, I've
got to make sure I'm putting them in the right spot.
So, if I have the hypotenuse, it's got to go where H is.
So, if I want to get B by itself, I need to move 15 squared to the other side of the equal
sign.
And since this B and 15 squared are separated by a plus sign,
I can just move it to the other side.
So, 22 squared is going to stay on the left.
This 15 squared is going to come over, it's on the left now as well.
But something that hopefully you [inaudible] out, crossing the equal sign with a value
is its sign changes.
There isn't a sign in front of 15 squared, but I can assume that that's positive.
If it was negative, there would be a negative sign there.
It was positive here, when I flip it over, now it's negative, don't forget that.
Because that'll change your answer significantly.
And then the B squared is sitting by itself. Now we're solving something pretty similar
to what we were doing when we were solving for the hypotenuse.
I need to get B by itself. It's on its own, but it has a squared above it.
I'd love to get rid of that, and the way I'm going to do that is take the square root of
B squared and those two will cancel themselves out.
But I've got to do the same thing to both sides of the equal sign.
So, I'm taking the square root of both.
This square root and the squared are gone.
So, B is by itself, which was great, that's what we're looking for.
Then I've got 22 squared, which I'm going to do quickly in my calculator here.
I might as well do that while we're here.
So 484... And then I'm going to do 15 squared, but I'm going to make sure that that negative,
that subtraction sign, stays a subtraction sign.
If you type in your calculator negative 15 squared, your calculator may give you a positive
answer but what we actually want is this value subtract this value.
So, I'm going to square the 15 with a positive 15.
And 15 squared is 225. Negative sign still intact.
You can skip steps here if you want to, but I'm not going to because I kind of want to
keep us all on the same page.
I'm going to do that subtraction first.
Remember that a square root sign is kind of like a bracket.
You've got to do everything underneath it first before you can actually press that square
root button on your calculator.
So, I'm going to do 448, subtract 225, sorry 484, I should probably write the right numbers
in there. And I get 259 equals B. And then I'm going to take the square root of that.
Depending on your calculator, you may have to type that square root button in first.
I don't need to. I could just type square root, if 259 is sitting on my screen.
And if I press equals, I'm getting 16.1. If I'm rounding to 1 decimal place. So, 16.1
equals B. And that's in metres, which means this would be in metres, too.
Want to make sure that if you know the unit that something's measured in, include it,
because 16.1 doesn't mean a whole lot. 16.1 metres, kilometres they mean different things.
So, make sure the unit's there.
Something we talked about in our last video that I'll mention again is the hypotenuse
is the side directly across from the biggest angle.
So, it should actually always be the biggest side. 22 is bigger than 16.1 so that makes sense.
It's kind of a good check for you.
So, what I'm going to do now is I am going to make myself some room here and I'm going
to come up with an application question for you to do on your own.
Okay, so let's take a look at this application.
This says that a rectangular foundation has one side that measures 36 inch-- or sorry,
36 feet, and a diagonal that measures 50.
So, before we finish reading that, any time you're doing an application question, if you
can, it's always a good idea to draw a diagram.
It makes it a little bit easier for you to understand what that question is asking.
Sometimes a paragraph of information can be really intimidating.
A diagram makes it a little bit easier on you.
And also, if I'm your client, maybe that's intimidating to me.
But if you can draw me a picture,
I get a better understanding of what you're talking about.
So, diagrams are good.
So, there I have a rectangular foundation.
And this says that one side has a measure of 36 feet.
And has a diagonal of 50 feet. A diagonal, now I could come from this side, down that
way, or from this side the other way.
They're both diagonals and if this is a rectangle, they should be the same.
I'm going to use this one, though.
So... this is 50 feet. So, there's our diagram.
A little bit easier to understand now.
And the question says how long should the other side be.
Now this is 36 and this is 36 as well.
How long should the other side be if the foundation is a true rectangle.
Now you might be looking at my diagram as I am right now, thinking mine doesn't look
like a true rectangle.
But when you're building, it's really important that your foundation is square.
You're building everything up from it, so it needs to be perfectly square.
So, this is a trick that's actually used on a pretty regular basis.
So, what we've got here is if this is a perfectly square rectangle, then both of these would
be right angles.
If they weren't, we're not perfectly square.
So now hopefully your brain's kind of flipping back to what we did a minute ago and you can
say, hey, there is actually a right triangle in there. And there is.
So, I'm going to try and pull that triangle out for you.
You don't necessarily have to do this but it's kind of nice to just see it.
So, there's my right angle right there.
If this is 36 feet, and the diagonal, which is our hypotenuse, is 50 feet, the question
is how long should that side be?
So hopefully you are sort of going yeah,
I think I can do that. Using our Pythagorean theorem that we have written over there.
So, I'm going to give you a minute to work through this on your own and then we'll come
back and we'll make sure that we're on the same page.
[ Music ]
Okay, so hopefully you gave that a shot.
Let's do it together, make sure we're on the same page.
Pythagorean theorem right there. So, I am going to write that down.
H squared equals A squared plus B squared and hopefully you know that the H value is
this 50.
It's like the most important part of the question.
If you don't put the hypotenuse in the right spot, you're not going to get the right answer.
So, I'm going to put 50 right here.
I am not going to forget the squared.
I am going to put my 36 here.
And then my other value that I don't know.
We could call it B. We could have called it A. Either way.
Now what we did with our other question was first we moved this value to the other side
of the equal sign, so let's do that.
I've got 50 squared subtract, because it's crossing the equal sign, 36 squared equals
B squared, which is the guy we're looking for.
And then let's get rid of that squared by putting a square root over the other side.
So, the square root of 50 squared, subtract 36 squared, equals B. By taking the square
root of this side, we could get rid of that squared at the B. And then I'm going to do
this in my calculator.
If you can do it in your head, fantastic, I can't, so I'm going to do it on my calculator.
So, 50 squared is 2500. I'm going to leave it under the square root sign, don't forget
that.
Let's make sure you can actually see my square root sign there.
So 2500... This subtraction sign stays a subtraction sign.
Now remember, don't do the square root of negative 36, because that'll flip to a positive
and that negative is important to the question. So, 36 squared is 1296 which equals B. And
my marker's having a hard time here.
So, my B value, I'll find if I first take 2500 and subtract 1296, so hopefully you got
1204 and that's the number I'm going to take the square root of.
So, I'm going to put my square root button on there. And I'm getting 34.7. Rounded.
And these were in feet, so this would be in feet as well.
And hopefully you can take a look at that and you can know that the hypotenuse should
be the longest side, so there's no way this should be longer than 50.
So that's a good check for you.
So hopefully that helps you understand Pythagorean theorem.
Hopefully you already understood how to find the hypotenuse from our other video, and now
hopefully that clears up what do I do if I'm not looking for the hypotenuse and I'm looking
for the other-- one of the other sides.
This is important stuff to know, especially if you're in a trade, this type of things
I used, the example I used is because it's used on a pretty regular basis in building.
So, it's worth knowing and hopefully that helps you understand it.