Lesson video

Hello, my name is Dr.

Rowlandson, and I am happy to be helping you with your learning during today's lesson.

Let's get started.

Welcome to today's lesson from the unit of geometrical properties and Pythagoras' theorem.

This lesson is called similarity in shapes, and by the end of today's lesson we'll be able to recognise that similar shapes have sides in proportion to each other, but the angle sizes are preserved.

Here are some previous keywords that you may remember, and we'll be using these words again in today's lesson.

So you might want to pause the video at this point while you remind yourselves the meanings of these words before pressing play to continue.

This lesson contains two learn cycles.

In the first learn cycle, we're going to be looking at shapes and deciding whether or not they are similar.

And then the second learn cycle, we're going to look at some more complex diagrams where there are multiple shapes within the same diagram and trying to find similar shapes within them.

But let's start off with identifying whether our shapes are similar.

Let's begin by remind ourselves some properties about similar shapes.

When one shape is an enlargement of another, they are similar.

For example, the pair of triangles on the left, they are similar.

And one way we could look at that is by looking at the multipliers that get us from the lengths of the small triangle to the lengths of the bigger triangle.

In each case, we are times them by two.

3 times 2 is 6.

9 times 2 is 18.

Because those multipliers are the same, it shows us that these triangles could be similar.

Whereas with the non-example on the right, we can see that the multipliers are not the same.

3 times 3 is 9, but 9 times 2 is 18.

The fact that those multipliers are different tells us that those triangles are not similar.

Also, when two shapes are similar, the lengths are in the same proportion.

With the example on the left, we look at those two triangles.

In both cases, the height is three times the base.

The height and base are in the same proportions.

So that tells us that those triangles could be similar.

But with the non-example on the right, we can see that for one triangle the height is three times the base, but for the other triangle, the height is two times the base.

They're not in the same proportion, so those triangles are not similar.

Here we have Aisha and Lucas.

Aisha and Lucas both draw a quadrilateral and they tell us some stuff about it.

Aisha says, "My quadrilateral has two lengths that are three centimetres and two lengths that are four centimetres." Lucas says, "My quadrilateral has two lengths that are six centimetres and two lengths that are eight centimetres." Now, Sam's listening in on this, and Sam says, "Hmm, Aisha and Lucas' quadrilaterals must be similar." But Jacob is also listening and Jacob says, "We can't tell if they are similar based on this information alone." Hmm, I wonder who you agree with, Sam or Jacob? Do you think that they must be similar, and agree with Sam? Or do you think we can't tell based on this information, and agree with Jacob? Pause the video while you think about this, and think about what your reasons are for your decision, and press play when you're ready to continue.

Let's hear from Jacob a little bit more and see what he means when he says that there's not enough information.

Jacob says, "You can't tell if two quadrilaterals are similar based on the lengths alone.

This is because different quadrilaterals can be made by arranging the same lengths in different orders." For example, here we have a rectangle, but if we just adjust these angles slightly, we get a parallelogram.

And we can see that these two shapes are not similar to each other.

Or another thing we could do is take those four edges and rearrange 'em in a different way.

Put the two short edges together and the two long edges together, and we have a kite.

And that is also not similar to the other two quadrilaterals.

So Sam's listened to this and Sam says, "Hmm, so does two shapes need to have the same angles in order to be similar?" When two shapes are similar, the angles are the same in each shape and the lengths are in the same proportion as well.

For example, here we have two parallelograms, and we can see that these are similar to each other by think about both of those properties.

For one, they have the same angles.

Yes, we can only see two of the angles that are labelled, but the markings in the vertices tell us that the other two angles are the same as the ones we can see.

We have 100 in the top left hand vertex of each parallelogram, and that has two arcs in both cases.

And the bottom right hand vertex of each parallelogram also has two arcs, that means they must be 100 as well.

Also, the length on these two parallelogram are in the same proportions.

If we divide 3 by 4 in the small parallelogram we get 3/4.

And if we divide 6 by 8 on the larger hologram, we also get 3/4.

That means 4 times 3/4 is 3, and 8 times 3/4 is 6.

These lengths are in the same proportions.

So the fact the angles are the same and they're in the same proportions shows us that these shapes are similar.

So let's take a look at a few non-examples.

Here we have two parallelograms. Can you see why these two parallelograms are not similar to each other? The reason why is because the angles are not the same.

We have 100 degrees and 80 degrees in the small parallelogram.

And we have 130 degrees and 50 degrees in this bigger parallelogram.

So they don't have the same angles.

Even though the sides are in the same proportion because the angles are not the same, they are not similar.

Here's another non-example.

Can you see why these two parallelogram are not similar to each other? Now they have the same angles, but the lengths are not in the same proportions.

4 times 3/4 makes 3 with a small parallelogram.

But with the big parallelogram, 10 times 3/5 make 6.

Those multipliers are different, therefore they are not in the same proportions and these shapes are not similar.

Here we have Sofia.

Sofia is trying to justify why the two shapes below are not similar.

Now we can see they're not similar, because they are different types of quadrilaterals.

We have a parallelogram and we have a trapezium.

But based on what we've talked about so far, we can see why Sofia's having a bit of a difficult time justifying why these two shapes are not similar.

Sofia says, "I can see that they are not similar, but they have the same angles.

Also the lengths shown are in the same proportions." So why are these shapes not similar? Pause the video while you think about this and press play when you're ready to continue.

Well, one way we can think about this is by considering the lengths.

We know that parallelogram has three centimetres and four centimetres, and those edges appear twice.

So the top horizontal edge must be four centimetres, and the slopey edge on the right must also be three centimetres.

With the trapezium, we know the eight centimetres at the bottom, we know the six centimetres on the left, and the edge of the right must also be six centimetres, but we don't know that top edge.

We can see it's smaller than eight centimetres.

But without that information, we might have to think about a slightly more robust way of explaining this.

One way could be to look at not just what the angles are, but what order the angles are in.

Sofia says, "While the two shapes have the same angles, they're not in the same order." Let's take a look at that.

If we call the top shape shape A and start on a top left vertex and go clockwise round, we have 100 degrees, 80 degrees, 100 degrees, and 80 degrees.

We're constantly alternating between 100 and 80.

Whereas if we call the shape at the bottom shape B and start on the top left vertex again, we'd have 100 degrees, 100 degrees, 80 degrees, and 80 degrees.

Those angles are not in the same order.

And one way we can see that is shape B has two consecutive angles that are 100 degrees, but shape A does not.

Same with the 80 degrees as well.

Even if we went anti-clockwise around shape B or started in a different vertex, the angles still would not be in the same order as with shape A.

So these two shapes are not similar.

That means when shapes are similar, the angles are preserved and in the same order.

Let's think about order of angles a little bit more with this example.

Here we have Andeep who is trying to decide whether the shapes below are similar.

We have two triangles.

They each have 37 degrees, 53 degrees, and a right angle.

Andeep says, "Does the order of angles matter with triangles? Each angle is connected to every other angle by an edge, so they are always in the same order." If you imagine, for example, three people, and they're all holding hands of each other, if they wanted to switch the order that they were in, they wouldn't be able to do it, because each person is holding hands with the other two people no matter which way round they go.

So they're always in the same order.

They might reverse clockwise, anti-clockwise, but they're always in the same order one way or another.

And that's what we've got going on with triangles.

There are three vertices, they're all connected to each other by edges, so whichever way round they go, they will be in the same order.

That means triangles with the same angles are always similar because the order can't be switched around like they can with quadrilaterals or shapes with more vertices.

So let's check what we've learned there.

True or false, if two shapes have the same angles, then they must be similar? Choose either true or false and a justification below.

Pause the video while you do it and press play when you're ready to continue.

The answer is false, because if shape has four or more vertices, then the angles may not be in the same order.

True or false, if two triangles have the same angles, then they are similar? Is that true or is it false? And choose a justification.

Pause the video while you do it and press play when you're ready to continue.

The answer is true, that's because every vertex in a triangle is connected to both the other vertices, so the order of the angles will always be the same.

True or false, if two shapes do not have the same angles, then they are not similar? Is that true or is it false? Pause the video while you think about it and press play when you ready to continue.

This one is true.

And the reason why is it is not possible for all the lengths to be in the same proportions if the angles are not the same.

And finally, true or false, regular shapes with the same number of sides are always similar? Is that statement true or is it false? Pause the video while you choose and then press play when you're ready to continue.

The answer is true, and that is the shapes will always have the same angles and the scalars between the sides within each shape will be one.

Another thing that we can work with when dealing with similar shapes is a ratio table.

A ratio table can help us find proportions between lengths within the shapes and find scale factors between similar shapes.

Let's look at that example again we saw earlier.

Here we have two similar parallelograms. If we put these lengths into a ratio table, like so, we can see here that the scale factors from shape A to shape B is times two, and we can also see that the proportions within the shapes are the same as well, times 3/4 each time.

Now this might not always be necessary, but it could be helpful for more complex cases.

If you've got lots of sides, and lots of different angles, and different orders, it might be a way to organise your information.

Let's put that to the test now.

Laura draws three similar rectangles.

She measures some of the lengths and widths, and fills them in on a ratio table.

What is the length of rectangle B? Pause the video while you write this down and press play When you're ready for an answer.

The answer is 24 centimetres.

So what is the width of rectangle C? Pause the video while you make a choice and press play when you're ready for an answer.

The answer is five.

Over to you now for task A.

This task contains two questions, and here is question one.

You need to decide are these two shapes below similar? But more importantly, you need to justify your answer with reasoning.

Pause the video while you work on this and press play when you're ready for question two.

And here is question two, Alex, Izzy, Jun, and Sam have drawn some quadrilaterals.

Alex says, "This is my quadrilateral." And you can see a diagram showing Alex's quadrilateral.

Izzy, Jun, and Sam all say some things about their quadrilaterals.

And what you need to do is sketch possible quadrilaterals for Izzy, Jun, and Sam.

Now when I says sketch, you can just draw some quadrilaterals and label the side and angles, and make it kind of look right.

Or if you want to go a bit further, you could measure these more accurately with a ruler and a protractor.

Whichever way you do it, pause the video and have a go at it, and then press play when you're ready for an answer.

Okay, let's see how we got on then.

Here's question one, are the two shapes similar? And we have to justify our answer with reasoning.

So we can see that each of these shapes have the same angles, but are those angles in the same order? Now, I always find it helpful to just label one shape, shape A, or another shape, shape B if there's no labels on there already.

But you might have went about it in a slightly different way.

If we call our left shape A and start in the top vertex, we have 72 degrees, and if we go clockwise round, we have then 127 degrees, 116 degrees, 90 degrees, and 135 degrees.

Can I create that same order of angles in shape B? Well, with shape B I want the first angle to be 72 degrees, that's in the bottom left vertex.

I want my next angle to be 127 degrees, I can see that if I go anti-clockwise round, so that's okay so far.

But then the next angle is 90 degrees rather than 116 degrees.

So with shape A, I can see that I have 127 followed by 116, but that's not the case in shape B.

Those two angles have a 90 degree angle in between them.

And also we can see with shape A I have a 90 degree angle followed by 135 degrees.

And in shape B, those two angles are not next to each driver.

So these angles are not in the same order, therefore these shapes are not similar.

And then question two, we need to sketch a possible quadrilateral for Izzy, Jun, and Sam.

Let's start off with Izzy.

So Izzy says, "My quadrilateral is similar to Alex's and all the edges are twice the length." That means I want all the angles to be the same, and I want to take the four centimetres and the six centimetres and times them both by two.

It'll look like this.

Here's Jun.

Jun says, "My quadrilateral is not similar to Alex's, even though all the edges are twice the length." So we know we want the edges to be 12 centimetres and eight centimetres, but how can we make it so it's not similar? Well, one way is we could use different angles and have a slightly different parallelogram.

Or another way is we could arrange those edges in a different order and have something like this, a kite.

And Sam.

Sam says, "My quadrilateral is not similar to Alex's, even though all the angles are the same." How could we do that then? We want another quadrilateral that has two 120 degree angles and two 60 degree angles, but we'd have to put 'em in a different order.

Here we are, we have a trapezium, same angles in a different order, and that means it's not similar to Alex's.

Well done so far.

Let's now move on to the second learn cycle, which is finding similar shapes in complex diagrams. Let's begin by remind ourselves about some of these key properties with similar shapes.

For starters, we know that similar shapes have the same angles.

But for triangles, the order of those angles doesn't matter because they're always in the same order.

So for triangles, when triangles have the same angles, they are similar.

So here we have two triangles and we want to check if they're similar, we could do it by comparing the angles of the small triangle with the angles of the bigger triangle.

We could cut the small triangle out or draw it on tracing paper and place it at each vertex in the bigger triangle and see that the angles are the same.

And because the angles were the same, we know that those triangles are similar.

So let's apply that now to this complex diagram here.

Well, here we have two triangles, a small triangle inside a bigger triangle.

ABE is our smaller triangle and ACD is our bigger triangle that has a smaller one inside it.

We're told that triangle ABE and triangle ACD are similar and in the same orientation.

How could this fact be used to show that the line segments BE and CD are parallel? Pause the video while you think about how we might go about this and then press play when you're ready to do it together.

Well, we could start by thinking about the angles in these similar shapes.

We know that angle ABE must be equal to angle ACD, because they are similar and in the same orientation.

And we also know that angle CDE must be equal to angle BEA for the same reason.

They are similar and in the same orientation.

So we have this pair of line segments, BE and CD, and we have in each case a transversal going through those points, B and C, and we have those angles there, ABE and ACD, which are equal to each other.

Those angles are corresponding with each other and they're equal.

And what we know is correspondent angles are equal when they're on parallel lines.

And we can put that same argument across, if we think of A to D being a transversal going through points E and D.

We can see that angles CDE and BEA are corresponding.

And because they're equal, it means that that pair of lines, BE and CD, they must be parallel.

So therefore BE and CD are parallel.

So we could turn that around now and use facts about parallel lines to show that two triangles are similar.

Here we have a complex diagram that looks a bit like the last one, where we've got a triangle inside a triangle.

And what we want to see is whether or not ABE is similar to ACD, the small triangle and the big triangle.

But what we can see to help us is that the line segments EB and DC are parallel, and we can see that because the arrows.

So let's go about proving those are similar.

Now we can see that both of these triangles contain the angle EAB, that's the angle right at the top vertex there.

And we can see that the angles in the bottom right vertex for each triangle, ABE and BCD, those are equal because they're on parallel lines and they're correspondent.

And we know that correspondent angles in parallel lines are equal.

And we can put the same argument across for the angles in the bottom left vertex of each triangle.

Angle CDE is equal to angle BEA, because they are correspondent with each other, and correspondent angles in parallel lines are equal.

Therefore, the two triangles, ABE and ACD have the same angles, therefore they are similar.

Here's a different looking diagram.

Prove that triangle ABE and BCD are similar.

And we have those line segments, EA and DC, as being parallel.

Maybe pause the video while you think about how we might go about this, what angle facts we might use in this case, and then press play when you're ready to continue.

Well, let's think about our facts about angles on parallel lines.

Here we have angle EAB and angle CDB, and they must be equal to each other, because alternate angles in parallel lines are equal.

And also angle BCD is equal to angle BEA for the same reason, alternate angles in parallel lines are equal.

And finally is the third angle in each triangle, ABE and DBC.

Now those are equal, and one reason we can give is that vertically opposite angles are equal, and we can see that the vertical opposites are at point B there.

Another way we could think about it is that there's only one angle remaining in those two triangles, so they must be the same if the other two angles in each of the triangles are the same as well.

Either way, triangle ABE and triangle BCD have the same angles, so they are similar.

Let's check what we've learned there.

Which angle is equal to angle CAB in this diagram? Pause the video while you choose either A, B, or C, and press play when you're ready for an answer.

The answer is A, the angle CDE is equal to angle CAB.

And the reason why is that they are alternate on parallel lines.

Which angle is equal to angle BCD? Your choices are A, B, and C, pause the video while you make a choice and press play when you're ready for an answer.

the answer is A, that's the angle ABE, and the reason why that they are equal is that correspondent angles on parallel lines are equal.

So now we know how to match up angles which are equal to each other on these complex diagram.

Let's use that to help us find some missing lengths when it comes to similar shapes.

Knowing which angles are equal to each other can help us identify corresponding lengths between similar triangles.

For example, here we have a complex diagram, and it says, find the length of AE, and all lengths are given in centimetres.

Perhaps pause a video and have a think about this yourself before press and play to continue together.

So what's tricky here is that the two triangles are in different orientations.

So working out which lengths correspond with which lengths can be a bit tricky.

But we can use the angles to help us work out which lengths must correspond with which.

For example, the edge AE is between an angle with one arc and an angle with two arcs, and so is the edge CD.

So you must know that those two edges correspond with each other.

So if I want to get the length of AE, I'm going to use that five and a scale factor to get to it.

So where's that scale factor will come from? Now you might be tempted to think that BC corresponds with AB, therefore the scale factor would be 12 divided by 4 is 3.

That's not an uncommon thing to think, but let's not be too hasty.

Because EB is between an angle with three arcs and an angle with two arcs, and so is BC.

That's between an angle with three arcs and an angle with two arcs.

So it must be those two edges that correspond with each other, and therefore the scale factor is 8 divided by 4 is 2.

So now we know the scale factor is two.

We can take the length of AE, five, and times it by 2 to get 10.

So our answer is 10 centimetres.

When solving problems involving nested similar triangles, that means a triangle inside another triangle, it can be helpful to draw the two triangles separately.

For example, here we have ABCDE, where we've got a small triangle, ABE, inside a big triangle, ACD, and we want to find the length of AD, and all lengths are given in centimetres.

So this might be a bit tricky to see with the diagram we have here.

And you might be tempted to think that the scale factor is 2 from doing 10 divided by 5.

That's not an uncommon thing to think.

But let's not be too hasty.

Let's draw these two triangles out separately.

Let's draw triangle ABE, which we have five and four centimetres.

Let's draw the bigger triangle now, ACD, it looks like this.

Let's think about that length from A to C.

To get from A to C on the original diagram, we go five centimetres across first and then another 10 centimetres across.

So it must be 15 centimetres altogether, therefore the scale factor must be 15 divided by 5, which is 3.

And now we know the scale factor is three, we can multiply 4 by 3 to get the length of AD, which is 12 centimetres.

Now, if we wanted to get the length of ED instead, well, we could do all the same stuff we've just done, but then just do another little calculation at the end.

Because we can see that the length of ED is equal to the length of AD, subtract the length of AE.

It's a difference between those two lengths.

So we do that, do 12 subtract 4, which gives us 8 centimetres.

So let's check what we've learned with that.

Which edge in the triangle CDE is an enlargement of the edge AC? Pause the video while you make a choice and press play when you're ready for an answer.

Well, the edge AC is between an angle with two arcs and an angle with three arcs, and that's also the case for the edge CD there.

Find the length of the edge CD.

Pause the video while you work this out and press play when you're ready for an answer.

Well, we can get our scale factored by dividing 12 by 6 to get 2, and then apply that to the 4 to get a length of eight centimetres.

Here we have another diagram.

EB is three centimetres long.

What is the length of DC? Pause the video while you work it out and press play when you're ready for an answer.

Now you might be tempted to get a scale factor of four here by doing 8 divided by 2, but remember if you draw these triangles separately and the bigger triangle will have a length of 10 centimetres from A to D, which means it's 10 divided by 2 to get a scale factor of five, and then apply that to our length of three centimetres and we get an answer of 15 centimetres.

Over to you now for task B.

This task contains two questions and here is question one.

In each case you've got a complex diagram and you need to find the unknown length in each diagram.

Pause the video while you work on this and press play when you're ready for question two.

Okay, here is question two, and it's a practical task this time.

Take a piece of paper, doesn't matter how big it is, it just needs to be a rectangle, and fold it as shown below.

Now when you fold it, it doesn't have to be exactly the same as what you can see here, it just needs to be a diagonal fold going from somewhere on that left edge to somewhere on the right edge.

And what you should see is quite a bit of that paper will overlap with each other, but there'll be some parts of the paper that won't overlap.

The paper that does not overlap should make four triangles.

And what I'd like you to do is prove that these four triangles are all similar.

Pause the video while you have a go at this and press play when you're ready to go through some answers.

Okay, let's see how we got on.

In question one, we need to find the value of each unknown.

A is 15 centimetres, B is 16 centimetres, C is 20 centimetres, and D is 4 centimetres.

And in question two we had to show that these four triangles were similar.

Now we don't have any length and we don't have any angles, but we need to somehow show they're similar.

And the way to do that is to show that the angles are the same regardless of what they are.

So let's work through this together.

We could start by labelling the right angles, 'cause the right angles came from the four corners of the piece of paper.

So we know with each of these four triangles is a right angle triangle.

Then let's think about the other two angles in the triangles.

We could label one of the angles X.

Doesn't matter which angle is, just your working will be a bit different depending on which one you label as X.

If we label this one as X, then we can start thinking about what are the expressions for the other angles around this diagram? Angles in a triangle sum to 180 degrees.

So that bottom left vertex of that triangle we can see here, that must be 180 subtract the 90 degree angle we had earlier, subtract the X, which is 90 subtract X.

If we think about this angle in this small triangle here, well, vertically opposite angles are equal to each other, which means that must also be X, it's equal to the first X we wrote.

And then the other angle in that triangle, well, once again, we could do angles in a triangle sum to 180 degrees and show that it must be 90 subtract X.

And then this angle is vertical opposite, so it must be the same as the one we've just done.

This angle here, well, we could do angles in a triangle sum to 180 degrees.

We'll do a slightly different calculation this time, but we'll end up with X as that angle there.

This angle is vertically opposite the one we've just done, so that must be X.

And finally angles in a triangle sum to 180 degrees.

We can work out that remaining angle is 90 subtract X.

We can see that all four triangles contain the same angles, therefore they must all be similar.

Fantastic work today.

There were some pretty tricky questions in that lesson, but hopefully we got on okay.

Let's summarise what we learned in this lesson.

Two shapes are only similar if the multiplicative relationship between correspondent edges is the same for every pair of correspondent edges.

And angles are invariant between similar shapes.

Now, similar shapes are not always easy to spot, especially when you've got a complex diagram with a similar shape inside a bigger similar shape.

It's not always easy to spot where those similar shapes are, but also what the lengths are as well.

But we can use the angles and angle facts to help us along the way.

Finally, a ratio table can help you find a scalar and functional multipliers in similar shapes if it's a particular complex one and you need to organise your information in particular.

Thank you much today, have a great day.