# Lesson video

In progress...

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

In today's lesson, we're going to look at surds and triangles.

All you need for this lesson is a pen and a paper.

So grab these and when you're ready, let's begin.

And to start today's lesson, I would like you to try this.

List some possible values of a and b in each case.

You have two right-angled triangles.

In each case you have been given one length, one side length, and two others are missing.

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

Okay, so for support, I want you to think about what we learned about Pythagoras' theorem.

We learned that, Pythagoras's theorem states, in a right-angled triangle, the sum of the squares of the shorter sides is equal to the square of the hypotenuse.

The hypotenuse being, the longest side in a triangle.

It's the one opposite the right angle.

If you think this helps you, to make a start, pause the video now and have a go at this.

So now let's have a look at this together.

So we can mark and correct your work.

Okay, the first one, I know that the square of the two shorter sides, so this square here and this square there, are going to be equal to the square of the longer side.

Longer side is five centimetres.

So the square of that is 25 centimetres squared.

Now, if these two here, two squares that I've drawn, are going to add up to 25.

Then one of them could be, let's say a squared could be 20.

And I can say that b squared can be five because now 20 plus the five add up to 25.

If a squared is 20, what would a be? Really good job, a would be the square root of 20.

What would b be? Excellent, b would be the square root of five.

But, I have more than one possible answer.

I wonder if any of you chose this one, a squared is equal to 16 and then made b squared nine.

This will tell me, if a squared is 16, then a must be four centimetres and therefore b must be three centimetres because it's the square root of nine.

Okay, and with the second one here, I have one of the shorter sides being five centimetres.

So I know that the square of that, is five squared equals 25.

Now I can choose any value I want for b, I'm going to choose number three.

So if b is three centimetres, then b squared must be three squared, which is nine centimetre squared.

Therefore a the hypotenuse, is going to be the sum of the two shorter sides, the square of the shorter sides.

So 25 plus the nine that gives me? Excellent, 34 centimetres squared.

And that is a squared, therefore, a must be? Good job, square root of 34.

You probably have chosen a number that is slightly different for b.

I could have gone with something else too.

Look at this one, what if I choose b to be 12, what would be the square of that? Really good, 12 squared equals 144.

And if b is 12, what would a now become? well a squared will be 25 plus the 144, which is? Good job, 169, and therefore a would be, square root of 169, which is? 13, really good job.

I wonder which numbers you have chosen.

And now we're going to move on to the main task of today's lesson, where we're going to try and calculate the lengths of the missing sides of right-angled triangles using Pythagoras.

The differences is this time we're going to try and leave our answers in surd form.

If the length of missing side is an irrational number, because then if we leave them in surd form they are more precise.

Let's have a look at the first one.

I want you to have a little think about the first triangle, Where would I start with? Really good job, I would start by identifying the hypotenuse here, and I would call it c.

You can you call it any letter, Really.

Okay, what can I write down now? Really good job, five squared, yeah plus eight squared is equal to c squared, really good.

Now what is 25 squared? Now what is five squared? 25 plus 64 is equal to c squared, add them up, really good, 89 equals c squared.

But I don't want c squared, I just want c, so c is the square root of 89.

And the unit is metre, really good job.

Now, we're leaving it in surd form because it's more precise rather than writing a non-terminating decimal.

Let's have a look at the second one.

Again, I've been given the two shorter sides, four metre and seven metres.

I know that this is the hypotenuse here, and I've called it c, it's opposite the right angle.

Let's start writing things down, what do I write first? Good job, four squared plus seven squared is equal to c squared, really good.

Next that is 16 plus 49 is equal to c squared, really good job, add them up, 65 is equal to c squared and don't forget to square root, Really good.

And that gives me square root of 65 equal to c.

And again, I'm leaving it in surd form because the square root of 65 is an irrational number.

Really good job.

Let's have a look at the next one.

I've been given one of the shorter sides this time, and one of the longer sides.

What do I need to do? I know that this 12 is actually c, is the hypotenuse, and I need to find this side so I can call it b very good job.

Have a little think, what do I need to write down? Excellent, one squared plus b squared equals to 12 squared, and next, really good, b squared is equal to 12 squared minus one squared.

What is 12 squared? Really good, b squared equals 144 minus one, which gives me b squared equals 243.

Remember, I don't want b squared I want b, b is square root of 143 centimetres.

And I leave it in surd form because it's more precise.

Really good job.

Okay, and now it's time for you to practise the independent task.

calculate the lengths of the missing sides in each triangle.

If the answer is an irrational number, leave it in surd form.

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

Okay, so to help you, you need to remember that a squared plus b squared is equal to c squared, according to Pythagoras' theorem.

The two shorter sides, a and b, if you square them, add them up, they'll give you the square of the longer side.

Look at the first triangle, and the first step is always to identify where the hypotenuse is and give it a letter.

We're going to give it c.

Look at the second one, where is the hypotenuse? It's always opposite the right angle.

Now that you know, which one is the longest side and which two are the shorter ones? Pause the video and see if you can have a go at this.

Okay, and now let's mark and correct your work.

The first one, four squared plus six squared is equal to c squared, excellent.

Next, that's 16 plus 36 is equal to c squared, add them up, that gives us 52 is equal to c squared and therefore c is? Really good job, square root of 52 is equal to c.

And we remember to write the units.

Really good job if you had this correct.

Next one, what do you write down? Really good, 30 plus 50 squared is equal to c squared.

So what is 30 squared? And what is 50 squared? Really good, 900 plus 2,500 is equal to c squared.

Add them Up 3,400 is equal to c squared.

And what do we need to remember now? Good job, squared root it, so the square root of 3,400 is equal to c, and we leave it in surd form.

Okay, next one, slightly different right? This time I have been given the hypotenuse as 20 centimetres and I need to find the shorter side then I call it b.

Okay, what do I need to write down? Good job, 14 squared plus b squared is equal to 20 squared.

Now I want to find b what do I need to write? b squared equals to 20 squared minus 14 squared, really good job.

Next b squared is equal to 400 minus 196.

Subtract them, that gives me b squared is equal to 204.

And the last part, the square root, b is equal to square root of 204 centimetres and we leave it in surd form, really good job.

Now, if you had these, the first all three correct, you should be really proud of yourself.

So the next two were a bit trickier.

So let's go through them together.

Okay, let's look at the next two.

Just because we have lengths that are surds, so surd lengths, it doesn't mean that we change the way we do things.

We still do things exactly the same.

Let's look at the first one.

We have two shorter sides.

One of them square root of six, and one of them has a length of square root of 20.

Same step, so let's just start by identifying the hypotenuse, opposite the right angle, it's c.

So now I know, that I need the square of the, two shorter sides.

The square here would be six because one of the sides is route six.

So the square would be six.

What about the other one? Good job, it would be 20.

If I draw a square, I don't have enough space to draw it here.

What do I need to write as my first line of the calculation? Excellent, six plus 20 is equal to c squared.

Now we have to pay attention to this and make sure that we're not writing six squared, okay? because the square of this shorter side here, square of root six is just six.

Therefore, now what do I have? 26 is equal to c squared and therefore really good job, c is equal to the square root of 26 centimetres.

And we're done, next one.

Let's start by identifying the sides.

So I have the hypotenuse here, given as root 15 and I have one shorter side being root six.

So the next one, I'm just going to call it an a.

What did I do the same thing.

If I try and draw a square there, what would it have? And if I draw a square here, what do you think that square would be? Okay, a squared plus six is equal to 15 well done, a squared is equal to 15 minus six, good job, a squared is equal to nine.

Now what about a? a is equal to square root of nine, really good.

Now we leave it in surd form if it's an irrational number.

Is the square root of nine an irrational number? It's not well done.

a is equal to three centimetres, and we're done.

Really good job, if you had this correct, if not, please make sure that you're correcting your work.

It's really, really important.

Okay, and this brings us to our explore task.

Tye forms a triangle by joining two right-angled triangles.

One has side lengths of six centimetres and eight centimetres.

What is the third side length? What could the side lengths of the other triangle be? What type of triangle can you make by sticking together two right-angled triangle? I would like you to pause the video and have a go at this.

Okay, to start with, I could look at the first the triangle that I have, which is this one here.

identify the hypotenuse and that is eight centimetres.

So well done if you've done that.

What can I do now? Really good, I can call this side X and say, I want to work it out or calculate the length of it.

How can I do this? Really good, x squared plus six squared is equal to eight squared.

How can I find x now? Good job, x squared is equal to eight squared minus six squared.

And what would that give me? X squared equal? Really good, 28.

And therefore x is square root of 28 and I leave it in surd form.

Really good job, well done.

Now that I know that this side is root 28, I can calculate a and b or I can work out what they could be.

Remember, there will be more than one answer.

Really good job, now let's look at the second part.

What types of triangles can you make by sticking together to right-angled triangles? Let me show you what I have.

These are two, stuck them together.

They made an? Really good job, an equilateral triangle.

I handle other two that I stuck together, and they made, an isosceles triangle.

And with this one here, with the question, we have two triangles and when we stuck them together, they made a? scalene triangle.

So you can make all sorts of triangles from two right-angled triangles by sticking them together.

It'll be a really good idea, for you to get a piece of paper and actually cut the triangles and stick them together, and then take a picture of your work and share it with your teacher, they would be very, very proud.

This brings us to the end of the lesson, some fantastic learning today.

So a huge well done to you.

You've done a great job.

Now I want to remind you to do the exit quiz and make sure you read the questions carefully and complete them.