Lesson video

Hello there and welcome to this lesson with me, Dr.

Saada.

In today's lesson, we're going to look at Pythagoras' theorem.

In particular we're going to look at Pythagorean triples.

All you need for today's lesson is a pen and a paper.

So pause the video and grab these, make sure that you're ready.

And when you are, let's begin.

To start today's lesson, I would like us to recap what we've learned in lesson number five.

So, Try this.

Use the numbers one to 10 to fill in the blanks and find the length of the hypotenuse.

Can you find any combinations that create an integer hypotenuse? Pause the video if you're feeling confident and have a go at this, if not, don't worry I'll give you some support.

Okay.

If you need some support, well here it is.

Remember what we learned in lesson four and five about Pythagoras' theorem.

We learned that the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.

For example, if we look here, I drew a square already for you.

So let's assume out of these numbers, I'm going to select a number.

Let's select it to be three.

That this side here is three units length, in length.

What's the square of that? The square of that is nine, square of three is nine.

Choose any number you want here, any.

I could go with, for example, six.

Then what would be the square of that? That's 36.

Now add these two up, 36 plus nine.

Excellent, it's 45.

Now what number would go here? It would have to be the square root of 45.

And in this case, 45, the square root of 45 is an irrational number.

So we leave it as it is.

Now, can you try and work this out so that each of these numbers in the squares here is an integer, a whole number out of these.

Pause the video and have a go.

Okay.

Let's have a look at what answers you managed to get.

So, what number did you have for the shorter sides? I had three here and a four.

Did you have that? Well done.

And for the hypotenuse, that's a five.

Really good job.

Well done.

That's one of the possible solutions.

Do we have another? Okay, let's check if it's the same as mine.

I have a six and eight for the two shorter sides and 10 for the longer one.

Excellent.

Good job.

Now I want you to notice the two possible solutions that we had.

One of them was three, four, five.

This is six, eight too.

Can you see the relationship between the numbers? Really good.

Well done.

Six is double the three, eight is double the four and 10 it's double the five.

And that's something we're going to look at in a minute.

Really good job.

Let's make a start.

Okay.

Let's have a look at this triangle here.

I have a right angle triangle here.

And I'm going to say that we're starting with side lengths of one, for the two shorter sides.

I have no idea what the hypotenuse is so I'm going to write a question mark, and I'm going to call it C.

Pythagoras' theorem tells us that one squared plus one squared is equal to C squared.

One squared plus one squared is two, is equal to C squared.

And therefore, if I want to find C, I want to find the length of that longer side.

I have to square root the two.

So C is equal to square root two.

Now, how do we get, or when do we get the hypotenuse nice whole numbers? So I don't want to have surds in there.

What do I need to do? What numbers would that, what numbers would give me whole numbers for the hypotenuse.

Let's have a look at this one.

So I have another triangle.

This time, I'm going to use length of three and four, for the two shorter sides.

And again, I don't know what the hypotenuse is, hopefully you can see something can already.

Now, Pythagoras' theorem tells us that three squared plus four squared equal to C squared.

Let's calculate this.

What would the number be now? Next line, what would I write down? Good job.

Nine plus 16 equals C squared and now, really good, 25 equals C squared.

And to get C, really good, I need to square root it.

And square root of 25 that is a rational number and that is, really good, five equals C.

So now I have the hypotenuse being five.

I have the shorter sides being three and four.

So I have the three sides here, nice whole numbers.

I definitely have the hypotenuse as an integer.

Okay.

What is so special about this triangle? This is a special right-angled triangle because all the side lengths are integers.

This is called a Pythagorean Triple that as in like the three numbers, the three side lengths are all integers, okay? And that's what we're looking at today.

Okay.

So, so far we have seen in the previous example that we had the Pythagorean triple, when we had side lengths of three, four, and five.

And in the Try now we also had that three, four, five.

We saw that it was whole numbers for the three side lengths.

And there was another possible solution for the Try now where we said six, eight, and 10 also work.

Then we said that there was a connection.

We doubled each number in that Pythagorean triple three, four, five, six, eight, 10.

And it actually worked.

Now let's look at another example where it will work.

I have a right-angle triangle here, and I'm going to use side lengths of five and 12 for the two shorter sides.

I don't know what the hypotenuse is, so let's work it out.

What does Pythagoras theorem tell me? Good job.

Five squared plus 12 squared equal to C squared.

Now what would be the second line of my calculation here? Really good.

25 Plus 144 equal C squared.

Add them up.

Excellent job.

169 equals C squared and therefore C will be? Really good, the square root of 169 and that is 13.

So I know that the hypotenuse is 13.

It's a whole number.

It's an integer.

And my, the two shorter sides of this triangle are also whole numbers.

So now I can say that the integers five, 12 and 13 are also called Pythagorean triple because five squared plus 12 squared is equal to 13 squared.

It works.

Okay? A triangle with side lengths, five centimetres, 12 centimetre and 13 centimetre is right-angled.

Okay? It can be metres it can be centimetres.

It can be millimetres.

It really doesn't matter as long as the numbers are five, the triple are five, 12, and 13.

Now, if I ask you to try and have a little think about what are the numbers would work, knowing that we have five, 12, 13, as a triple that works.

What would be another Pythagorean triple? Thinking about what I said about the Try now.

Yeah.

Excellent.

So if I have five, 12, 13 and it works and if I double each side 10, 24, 26 should also work should be also a Pythagorean triple.

How do I check that this is a Pythagorean triple? I will need to do the calculation.

The two shorter sides, add them up.

Well, add the square of the two shorter sides, 10 square plus 24.

And check.

Does that equal to 26 squared? And if you plug this into your calculator, you will see that it does work.

So we know that a triangle that has 10, 24, 26 is actually a Pythagorean triple.

Okay.

And there we go.

Now it's your turn to independently practise these questions.

They're very similar to the questions I showed you earlier.

Let's read the question together first.

Use Pythagoras' theorem to determine which of the sets of numbers below are Pythagorean triples.

So you're going to check if these triangles or triangles with these side lengths, are they Pythagorean triples or not? If you're not too sure, you can go back to the previous example where I showed you how to do that.

Okay.

Pause the video and have a go.

Good job.

Let's make a start.

I'm going to go through the answers.

And I would like you to mark and correct your work as we go along.

So, to check I need to ask myself for the first one is six squared plus eight squared equal to 10 squared? 36 plus 64 is equal to 100 therefore yes.

Six, eight, 10 is a Pythagorean triple.

Really good job if you had that correct.

Let's check the next one.

Nine, 12, and 20.

Is nine squared plus 12 squared equal to 20 squared? 81 plus 144 does not equal 400.

So this time it does not work, therefore no.

Nine, 12, 20 is not a Pythagorean triple.

Really good job if you had this correct.

Now, if you've made a mistake, you can pause the video here and try C, D and E again, that you have seen how to work out a and B.

Otherwise, keep going.

Next one, 11, 22 and 33.

What did you write down? 11 squared plus 22 squared is equal to 33 squared.

Let's check.

What were your numbers like? Good job.

121 plus 484 does not equal 1089.

Really good job.

And therefore no, 11, 22,33 is not a Pythagorean triple.

Really good job.

Next one, 10, 24 and 26.

We looked at something like this before.

Didn't we? Yes, we did.

Really good job.

So, 10 squared plus 24 squared equal to 26 squared.

100 plus 576 is equal to 676.

That is correct.

So yes, 10 24, 26 is a Pythagorean triple.

And really good if you spotted that we've done something like this before where we looked at the triangle five, 12, and 13, because that, this one is a double of everything that we had seen in the previous one.

Last one.

What did you get? Really good.

Seven squared plus 24 squared equal to 25 squared.

Well, yes it does.

'Cause 49 plus 576 is equal to 625.

So yes, seven 24, 25 is a Pythagorean triple.

Really good job.

Well done, you should be proud of yourself.

And this brings us to our Explore task.

The diagram is made out of right-angled triangles.

What value is x? And for the second part, are each of the pink angles, the same size? If you're up for a challenge and you're really confident, you can pause the video now and have a go at this.

If not, don't worry, I'll give you some support.

Okay.

Let's have a look at this.

Let's start by looking at the first triangle that I have here.

It's a right angle triangle.

And 12 centimetres is the hypotenuse of this triangle.

I can call this side anything I want.

I'm going to call it A.

What does the Pythagoras' theorem tell me? It tells me that A squared plus four squared is equal to 12 squared.

Excellent.

Now, how can I find A squared? Really good job.

A squared is equal to 12 squared minus that four squared.

Let's calculate this.

A squared is equal to 144 minus 16.

Let's do the subtraction.

A squared is equal to 128.

Now, our A squared is 128.

I can go and say, well therefore A is equal to the square root of 128 and leave it like this, in surd form because it's an irrational number.

But I don't really need to know what A is, 'cause what I'm going to do now is I'm going to try and find the next side, which is here.

I'm going to try and find this one.

I'm going to call it B.

And to do that, I will have to square A anyway.

So I may as well have just kept it as A squared equal to 128.

Okay.

So now I can say that B squared plus four squared is equal to A squared.

I wrote it here as square root of 128.

Just in case you have simplified in the first step.

Okay.

Now let's find B squared.

B squared is going to be 128 minus 16.

Therefore B squared is 112.

I'm not going to write down what B is because I will have to square it again anyway.

So I may as well just carry on.

Now, I'm going to label this side C.

I would like you to pause the video here and have a go at this.

Find C.

Once you've found C try and use the same method to find X and then press play again.

Really good job.

If you tried to work out C, C squared is 96.

Again, we don't need to find C and to find X.

We need to do exactly the same thing.

X squared is 80.

But this time, we aren't trying to find the value of X.

We're not going to use it for anything else.

We want X.

So X is the square root of 80 centimetres.

And I've left it in surd form because it's more accurate, more precise than having a decimal that does not terminate.

Really good job if you had this correct.

Now, let's have a look at the second part of the question.

Are each of the pink angles the same size? Now, each of these four triangles have a side length of four centimetres, but are all of their hypotenuses the same? They're different, aren't they? So, not all the sides are the same.

One side is the same, not all of them.

What does that tell us about the angles? Yeah, you're right.

They're not going to be the same and not the same size.

And in fact, this is a topic that you're going to learn about later on in this year.

I cannot believe that this is the end of the lesson.

We've done so much work today.

Well done.

You should be so proud of yourself.

Now, I would like you to do three things for me.

The first thing, I would like you to have a look at your notes and the examples and the questions that you wrote down in your book.

And I want you to identify three, the three most important things that you learned from today.

They could be anything that you've picked up from today's lesson, it's entirely up to you.

Second thing, I would like you to do the exit quiz.

I would like to see what kind of scores you get.

So please have a go at that.

And lastly, if you would like to share your work with Oak National and would like me to have a look at it, please ask your parent or carer to share your work on Twitter, tagging @OakNational and #LearnwithOak.

And that is it from me today.

Thank you very much for listening today and for doing such an amazing amount of work.

I'll see you next lesson.

Bye.