Lesson video

Hello, I'm Mrs. Lashley and I'm going to work with you as we go through the lesson today.

I really hope you're looking forward to the lesson.

You're willing to give it your best shot and I will be there to support you as we get through it.

So our learning outcome today is to be able to use Pythagoras' theorem to justify whether a triangle is right angled.

There are some keywords that I'll be using during the lesson, you've met them before, but let's just have a look at them again.

So the hypotenuse is the side of a right angled triangle, which is opposite the right angle.

And Pythagoras' theorem states that the sum of the squares of the two shorter sides of a right angle triangle is equal to the square of the hypotonus.

And as we just said, the hypotonus is that edge opposite the right angle.

It's also the longest edge.

So our lesson is gonna be broken into two learning cycles so that we can use Pythagoras' theorem to justify a right angle triangle.

And so on the first learning cycle, that's exactly what we'll be doing.

We'll be justifying right angle triangles and be using Pythagoras' theorem to do that.

When we get to the second learning cycle, we're gonna look at approximation using decimals.

So let's make a start at looking at how we make use of Pythagoras' theorem in order to justify a right angle triangle.

So we've met Pythagoras' theorem before.

And it states if any triangle is a right angle triangle, then the following is true.

So we can see the diagram there.

We've got our right angle mark, so we know that this is a right angle triangle.

And then if we have these squares drawn on each of the edges on the triangle, then what is Pythagoras' theorem? Didn't you remember? Well, it's A squared plus B squared equals C squared.

So the square of the two shorter sides, the sum of the square of the two shorter sides is equal to the square of the hypotonus.

And C is the hypotonus because it's opposite the right angle in a right angle triangle.

And in fact that this theorem is only true for right angle triangles.

So if you have a look on the left hand diagram, that is our diagram that we are familiar with for Pythagoras' theorem.

And it holds true A squared plus B squared equals C squared because it is a right angle triangle.

Whereas on the right hand side, that diagram where we've got squares along each edge, it is not true that D squared plus E squared is equal to F squared.

And that's because F is not a hypotenuse.

It's not a hypotenuse because it's not a right angle triangle and that relationship therefore does not hold.

So Jim says, "If all right angle triangles have the rule A squared plus B squared equals C squared and no other triangles have property then." And Izzy finished his sentence.

If A squared plus B squared equals C squared, it must be a right angle triangle.

So if that relationship holds then it must be a right angle triangle because it doesn't hold for any other type of triangle.

So this is known as the converse of Pythagoras' theorem and the converse of Pythagoras' theorem is if a triangle has the sides A, B, C, such that that relationship A squared plus B squared equals C squared, then it is a right angle triangle.

So A squared, B squared and C squared, if A squared plus B squared equals C squared.

So if that is true then this would be a right angle triangle because it only is true for right angled triangles.

So which of these triangles must be a right angled triangle? So behind the door there is contestant number one.

And all the contestant number one is telling us is its edge lengths.

So here we have, I have side lengths of four centimetres, seven centimetres and 10 centimetres.

So is it a right Angle triangle? So have a think about that for a moment.

Pause the video.

Would a triangle with the edges four, seven and 10 have a right angle? Would it be a right angle triangle? Izzy says this isn't a right angle triangle because four squared plus seven squared isn't a 100.

So where did the 100 come from? Well, 10 is the longest edge length.

So if this is a right angle triangle, that's the hypotheses.

10 squared equals 100.

4 squared is 16, seven squared is 49.

What is 16 plus 49? Well 16 plus 49 is 65.

65 is not a 100.

So Pythagoras' theorem does not hold.

Hence this is not the triangle we are looking for.

It doesn't have a right angle, it's not a right angle triangle.

So let's have a look at contestant number two.

Contestant number two says I've got side lengths of eight centimetres, 11 centimetres and 13 centimetres.

So Izzy's his thinking about this one, again, pause the video.

Do you think this is a right angle triangle or not? Press play when you're ready to move on.

This one isn't a right angle triangle.

If we do eight squared, you get 64.

If you do 11 squared, you get a 121, 64 add 121 is 185.

And what is 13 squared? Well 13 squared is 169 and 185 is not equal to 169.

Therefore this is not a right angle triangle.

So let's have a look at contestant three and this one is a check for you.

So pause the video and decide whether behind the door there is a right angle triangle or not.

Press play when you're ready to check.

This one is a right angle triangle, Pythagoras' theorem does hold for those three values.

Those three values were integer values and so actually was a Pythagorean Triple.

So we're up to the first task of the lesson for you.

And for this question, for each shape you need to calculate whether they are right angle triangles or not.

So you've got six triangles there.

You've got their lengths of the edges.

You need to decide if it's a right angle triangle.

You need to justify by using Pythagoras' theorem.

Remember if it holds then it is a right angle triangle.

If it doesn't hold then it is not.

So press pause and when you're ready for the answers press play.

So C, D and E, were right angle triangles.

Pythagoras' theorem did hold it was true, but for the others it was not.

So let's have a look.

C, we've got a five centimetre edge, a 12 centimetre edge, and a 13 seven centimetre edge.

So first of all, which one's the longest length? That's gonna be the potential hypotenuse.

So we know it is a hypotenuse 'cause this one is right angled, so 13.

So now we need to check does the sum of the square of the two shorter sides equal the square of that hypothesis? So five squared is 25, 12 squared is 144, and 13 squared is 169.

25 add 144 is also 169.

And that's why we can say that Pythagoras' theorem is true for those three values.

So this is a Pythagorean triple.

If we look at D, we've got 11, 60 and 61.

61 is our hypotenuse, it's the longest length and 11 squared plus 60 squared is equal to 61 squared.

And then if we look at E, we've got 24, 10 and 26.

So we've got 24 squared plus 10 squared does equal 26 squared.

The relationship Pythagoras' theorem.

A squared plus B squared equals C squared holds true.

And that's why we can justify this to be a right angle triangle.

Did you notice that E and C were enlargements? They were similar triangles.

If you have a look, we've got a five centimetre on triangle C and we have a 10 centimetre on triangle E, it was twice as long and there's a 12 centimetres and if you double that, you get the 24.

And then we've got the hypotenuse is 13 and on triangle E, the hypotenuse is 26, which is also double.

So this is a scaled up version of the right angle triangle, five, 12, 13.

We're now at the second learning cycle where we're gonna continue justifying with right angle triangles, but this time looking at approximations using decimals.

So Jacob asks, "Where might I use this property?" Izzy said, "It comes in handy when fitting windows and furniture and old houses." And we're gonna investigate that a little bit more, but pause the video and have a think for yourself.

Why might being able to prove that a triangle is right angled come in handy.

Press play when you're ready to move on.

So Izzy has pointed out one which we're gonna investigate.

And often it's to do with construction that this is the reason it's becomes most handy.

That if you need to build something and make sure there is a right angle, then using a triangle, knowing the three edge lengths of a right angle triangle is a really good way of making sure two things are square or perpendicular.

Two edges of a table, of a frame, two walls, et cetera.

So knowing that they are at right angle, knowing that they are perpendicular is very important in construction.

So how might we check if the edges for the window frame meet at right angles? So here we've got an image of a window frame that looks to be a nice sash window.

So how would we check that this frame is still square? That it's meeting at a right angle, that it's rectangular? Well if we know the width we could measure that with a tape measure and we know the height of the frame.

Then can we check that those two are perpendicular? Well if we also measure the diagonal distance, this would be our hypotonus if this is a right angle triangle.

So in everyday life accuracy to one decimal place is usually sufficient.

So we've now got some measurements of our frame.

So is the corner of the frame a right angle? Is the window frame square? So we are gonna use Pythagoras' theorem.

And if Pythagoras' theorem holds true, then yes, this is a right angle triangle and therefore the two edges are meeting at a right angle.

So let's have a look.

The square root of 46.

2 squared plus 30.

5 squared is equal to 55.

4 to one decimal place.

So the Pythagorean theorem works with these lengths.

So the triangle is a right angle triangle and therefore this frame, this window frame is rectangular, it's nice and square.

So here's a check for you.

Aisha has bought an exciting new cupboard and wants to know if it will fit in the corner of the room.

It will only fit if the two walls are perpendicular to each other, otherwise there will be a gap.

So have a look at the diagram.

So you've got two walls there.

One wall is 180 centimetres long, the other is 206 centimetres long.

We also have the diagonal distance from one end of the wall to the other end of the other wall.

And the orange rectangle is a plan view of the cupboard.

So will it fit into the corner of this room? Pause the video, and when you're ready to check, press play.

So Pythagoras' theorem, on those three measurements.

We're gonna apply it to the two shorter lengths.

And if the hypotenuse length is equal to this diagonal length, then we know that that is a right angle.

So square root of 206 squared plus 180 squared is 273.

6 to one decimal place.

So will the cupboard fit? Is that a right angle? Aisha says the walls do not meet at a right angle, so the cupboard will not fit exactly because if you look at the measured diagonal, it was shorter than the calculated hypotonus.

So it's not a right angle.

Those two walls are not perpendicular to each other, which means if this cupboard is very square, has got right angles, then it will have to have a gap.

So here we're up to the last task of the lesson.

And so for question one, you've got three door frames and you need to decide which doors have edges that meet at a right angle.

So you can see the three given measures.

You've got the width of the door, the height of the door, and the diagonal distance of the door.

So you are checking in this question whether they have been made square, whether they are rectangular in their shape.

So pause the video and decides which of those three, maybe all of them, maybe a couple of them are at a right angle.

Press play when you're ready for the next question.

So on this question we've got three triangles.

We've got all three edges of the triangle labelled up.

And the question is, which of these triangles have a perpendicular height of three? So if you look, all three of them have got at least one edge that is marked as three.

And is that the perpendicular height of the triangle? We know that perpendicular heights of triangles do not need to be the edge of a triangle.

And actually that's only the case when it is a right angle triangle.

Otherwise the perpendicular height could be labelled inside or outside to depending on the particular triangle.

So the question here is, is this a right angle triangle? Pause the video and decide if A, B or C, are right angle triangles.

And therefore have a perpendicular height of three.

Press play when you're ready to go through our answers to task B.

So here's our question one again.

So remember you were checking to see if these doors have got edges that meet at a right angle.

So on A, you needed to see if the edges 1.

1, 2.

3, and 2.

7 make a right angle triangle, whether they hold in Pythagoras' theorem.

And if we substitute the two shorter lengths and find the square of each of those and sum them and then square root them to give us the length of the hypotenuse, we can see that 2.

5 does not match the diagonal length of the door.

Whereas if we go to B, and we apply Pythagoras' theorem, the hypotenuse that is calculated is 2.

4 and that is the distance on the door.

So this one is at right angles.

And if we look at C, we've got 0.

8, 1.

9 and 2.

1.

Do those three values fit in Pythagoras' theorem? Does it make Pythagoras' theorem hold true? Yes.

So we should have just used Pythagoras' theorem to justify or to check if there is a right angle in that triangle.

Onto question two, as I said, you needed to see if there was a perpendicular height of three on the triangle.

The only time there is a perpendicular height of three that is also an edge is if it's a right angle triangle.

So does Pythagoras' theorem hold true? On A it does, at least to one decimal place.

Three squared plus three squared, and then square rooted.

On B, it does not.

Three, four and five is a very famous Pythagorean triple that is guaranteed to be a right angle triangle.

So here we've got two shorter edges of three and four.

So if it was to be a right angle triangle, then the hypotonus, the longest edge would be five exactly.

This is longer than that.

So that angle is not 90 degrees, it would actually be more than 90.

It'd be an obtuse angle.

And on C to one decimal place, this is true, it has got right angle, three and five are perpendicular edges.

It is a right angle triangle.

So to summarise today's lesson, which was used in Pythagoras' theorem to justify a right angle triangle.

For a triangle to be right angled Pythagoras' theorem must hold.

Because Pythagoras' theorem only holds true for a right angle triangle.

By substituting in the three lengths you can show whether the theorem holds.

Whether the square of the two shorter sides added together is equal to the square of the longest side.

If the three numbers produce a true statement, then the triangle is right angled, but if they do not produce a true statement, the triangle is not right angled.

So it's a way of justifying whether a triangle is right angled or not.

Well done today and I look forward to working with you again in the future.