Lesson video

Hello, my name is Dr.

Rowlandson and I'm excited to be guiding you through 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 Checking Your Understanding of Congruence.

And by the end of today's lesson, we'll be able to translate, reflect, and rotate a shape.

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

So you may wanna pause a video at this point if you need to remind yourselves about any of these words and press Play when you're ready to continue.

This lesson contains two learn cycles.

In the first learn cycle, we'll be looking at transformations that produce congruent shapes.

And in the second learn cycle, we will be identifying misinformation on congruent shapes.

But let's start off with transformations with congruence.

Here we have a paper rectangle that is placed on a desk.

Andeep, Izzy, Jacob, Laura, and Sofia discuss what they could now do with this shape.

Andeep says we could move the rectangle like this, and this would be a translation.

Izzy says we could flip the rectangle like this, and this would be a reflection.

Jacob says we could turn the rectangle like this, and this would be a rotation.

Laura says we could cut it into a smaller, similar rectangle, a bit like this, and this would be an enlargement.

Sofia says we could cut it into a different shape.

Now, this isn't a standard transformation, but it could look something like this.

So, we've got five actions there, four of them are standard transformations and one of them is something a bit different.

Andeep considers what effect each action has on the shape.

He says some actions change the lengths and some change the angles.

But can you think of which actions preserve both the lengths and the angles? In other words, for which actions do both the lengths and angles remain the same? Pause the video while you think about this and press Play when you're ready to continue.

The actions that preserve both the lengths and the angles are three of the transformations, translation, reflection, and rotation.

Shapes which have the same lengths and angles are congruent.

For example, on the left here, we have pairs of congruent shapes.

You can see that they're congruent because they have the same length and same angles.

However, on the right we have pairs of incongruent shapes, incongruent meaning they are not congruent.

We can see that our top pair of shapes are not congruent, or they are incongruent, because the lengths are different.

And for the bottom pair, well, they're different shapes to start with, but we can also see that some of the angles are different there as well.

So they are incongruent.

Izzy says this means that translation, reflection, and rotation always produce an image that is congruent to the object because the angles and lengths remain the same when you do those transformations.

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

True or false, the two shapes here are congruent? Is that true or is it false? Make a decision and choose a justification as well.

Pause the video and then Press play when you're ready to continue.

The answer is false.

Although they have the same lengths, their angles are different, so that means they are not congruent, they are incongruent.

True or false? These two shapes are congruent.

Pause the video while you make a choice and choose a justification, and then press Play when you're ready to continue.

The answer is false again.

Even though these have the same angles, the lengths are different, so they are incongruent.

And true or false, these two shapes are congruent? Pause the video while make a choice and choose a justification, and press Play when you're ready to continue.

The answer is true, even though they're in different orientations, that's fine when it comes to congruence.

The shapes have the same lengths and same angles, therefore, they are congruent, which transformations always result in an image that is congruent to the original object? Your choices are enlargement, reflection, rotation, and translation.

Pause the video while you make your choices and press Play when you're ready to continue.

The answer is reflection, rotation, and translation always results in an image that is congruent to its original object.

So now we know that translation, reflection, and rotation produce an image that is congruent to its object.

Let's practise doing those things together.

I'm gonna do a question on the left and I'm gonna give you one which is very similar to do on the right.

Produce a congruent shape by translating the object by the vector four minus three.

Let's do that together.

Remember, the top number in the vector, that tells us how far across we move with positive meaning to the right and negative meaning to the left, and the bottom number in the vector tells us how to move it vertically with positive numbers meaning moving up and negative numbers meaning moving down.

So let's take one vertex at a time.

This top left vertex, if we move it four right and three down, it goes here.

And the bottom right vertex, if we move that four right and three down, it goes here.

And then the bottom left vertex, if we translate that four right and three down, it goes here.

We have our three vertices for our image.

Let's join them up.

Here's one for you to try.

Produce a congruent shape by translating the object by the vector minus two, two.

Now, if you have a printed copy of this example, you can do it on that.

Otherwise, you can take a piece of square paper, draw the triangle on it with the same measurements, and then do the transformation on that.

Whichever you do it, pause the video, have a go, and press Play when you're ready to go through it.

Okay, your congruent image should look something like this.

It's been moved left two and up two.

And here's another example.

Produce a congruent shape by reflecting the object over the line.

Let's do this one vertex at a time.

The top left vertex, we can see it's four squares away from the line.

So its correspondent vertex on the image will also be four squares away from the line.

The bottom right vertex is two squares away, so its corresponding vertex will also be two squares away.

And then the bottom left vertex, that is four squares away, so it should also be four squares away on the other side of the line.

And we can also see that last vertex, but it's in line with the other two vertices that we plotted either below the first one we did , or the right of the second one we did.

So then when we join them up, we get our image looking like this, which is congruent to the object.

So here's one for you to try.

Take that triangle, produce a congruent shape by reflecting it over the line.

Pause the video while you have a go and press Play when you're ready to go through it.

And our answer should look something like this.

And then a third and final example, produce a congruent shape by rotating the object 90 degrees around the point.

Let's use tracing paper for this.

Place a tracing paper over the object and trace it.

Put our pencil on the centre of rotation so we keep that point fixed, and then turn the tracing paper 90 degrees clockwise.

Once we've done that, we just need to then reproduce our image onto the grid, and it looks like this.

The image is congruent to the original object.

Although it's in a different orientation, that's fine.

Here's one for you to try.

Produce a congruent shape by rotating the object 90 degrees anticlockwise around the point.

Pause the video while you do this and press Play when you're ready to check your answer.

Once you've done that, your answer should look something a bit like this.

Okay, it's over to you now for task a.

This task contains one question, and here it is.

We have seven shapes on the grid here, and parts a to g all give you an instruction for what to do with one of those shapes.

They'll tell you a transformation and how to apply that transformation.

Apply each transformation to each of the shapes as indicated, and it will produce an overall image.

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

Okay, let's see how we got on with that.

Once you perform all of these transformations, you should have a final result that looks something like this.

Check yours against the one on the screen, and if yours is different, don't worry about it.

Just try and find a particular shape which looks like it's in the wrong place.

Have another go at that particular transformation and see if you can get it right a second time.

Great work so far.

Now let's move on to the second learn cycle, which is identifying missing information on congruent shapes.

Reflection, rotation, and translation always produce an image that is congruent to the object.

When the two shapes are congruent, it means their lengths and angles are the same.

Now, the positions and the orientation of the lengths and angles may differ, but they will still maintain the same order.

So let's think about these points by working through a series of examples.

I'm gonna do a question on the left and then I'm gonna give you a very similar question to have a go at on the right.

Shapes A and B are congruent.

What transformation maps A onto B? Well, we can see that these two shapes are in the same orientation.

In fact, if I traced over A and I wanted to put the tracing paper perfectly on top of shape B, I could do that just by moving the tracing paper to the right.

So, in that case, this is a translation.

All of the points in shape A have been translated to the right to get to the same corresponding points on shape B.

Label shape B with the missing lengths and angles.

Now, this particular question is fairly straightforward because the shapes are in the same orientation, but even though that's the case, let's be systematically about our approach so we can apply then the same process to more difficult questions later on.

Let's start by looking at the angles.

Two of the angles in shape B are already given to us, the right angles.

Then let's look in shape A.

The largest angle in shape A is 118 degrees.

Let's find that in shape B.

It's here.

And the smallest angle in shape A is 62 degrees.

Let's find it in shape B.

It's here.

And let's just check that these angles are in the same order.

In shape A, I can read off these angles going clockwise around the shape by saying 118 degrees, 62 degrees, 90 degrees, and 90 degrees.

And in shape B, I can do exactly the same.

Start at the top right vertex, 118 degrees, 62 degrees, 90 degrees, and 90 degrees.

They're in the same order.

Let's now do the sides.

The shorter side on shape A is five, the shortest side on shape B is also five.

And then going around clockwise, we've got 17, 13, and 15.

Once again, those lengths are in the same order as they are in shape A.

Here's one for you to try.

Shape C and D are congruent.

What is the transformation that maps C onto D? And label shape D with its missing lengths and angles.

Pause the video while you do this and press Play when you're ready to continue.

So this is a translation and we can see that because they're in the same orientation.

And when we label shape D with it's missing lengths and angles, you should have these.

Here's another example, shape A and shape B are congruent.

What is a transformation that maps A onto B? Well, we can see they're not on the same orientation, so it's not translation.

If you imagine tracing over shape A, you could flip that tracing paper over and place it on shape B.

So that means it is a reflection.

And then label shape B with its missing length and angles.

Let's start off again with the angles.

The largest angle on shape A is 118.

The largest angle in shape B is 118.

The smallest angle is 62.

And then let's just check the order.

In shape A, I can read off 118, 62, 90, and 90 going clockwise.

And in shape B, I have to go the other way around, but I can still read them out in the same order.

118, 62, 90, and 90.

And now let's look at the lengths.

The length of the shortest edge on shape B is five.

So, on shape B, it must also be five.

And then when we look at the two edges that are adjacent to that five.

Well, one of them is 15, the other one is 17, 17 being the longest edge and 15 being the next longest.

Let's look at those in shape B.

The next one around going clockwise in shape B, that must be the 15.

And then if we look at shape A, if we go anti-clockwise, round those edges, you've got five, 15, the next one must be 13 and then 17.

Let's do that on shape B, but we'll have to go clockwise round.

So five 15, next one must be 13 and then 17.

Here's one for you to try.

What transformation map C onto D? And then label the shape D with its missing lengths and angles.

Pause the video while you have a go and press Play when you're ready to continue.

Once again, this is a reflection and when you label shape D with its missing lengths and angles, you get this.

And here's a third and final example to go through together.

Shapes A and B are congruent.

What transformation maps A onto B? Well, they're not on the same orientation, so we know it's not translation.

If we took some tracing paper and traced over shape A, we could place that perfectly on top of shape B by turning the tracing paper around.

We could rotate that shape 90 degrees clockwise to get shape B.

So it's rotation.

And label shape B with its missing length and angles.

Let's be systematic again.

Shape A, the largest angle is 118.

Let's find that in shape B.

It's here.

In shape A, the smallest angle is 62 degrees.

Let's find that in shape B, it's here.

And then let's double check the order.

So with shape A, if I go 118, 62, 90, and 90, can I do that in shape B by going one way around or the other? Yes, I can.

118, 62, 90, 90 going clockwise around that shape.

And then for the lengths, let's start that top length in shape B.

We can see it's between two right angles.

Which length and shape A is between two right angles? It's the 15, so it must be 15.

And then the next length around clockwise in shape B is the shortest length, so that must be five.

And then we can follow the same order around, 15, five.

The next must be 17, and then the last one must be 13.

And here's one for you to try.

What transformation maps C onto D? And then label shape D with its missing length and angles.

Pause the video while you have go and press Play when you're ready for an answer.

It is a rotation.

It's a rotation of 180 degrees, and when you label its missing lengths and angles, here's what you get.

Okay, it's over to you now for task B.

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

You've got five triangles.

All five triangles are congruent.

For part a to e, you need to identify which transformation maps A onto B, or B onto C, and so on.

And then for part f, you need to go back then and label the missing length and angles on triangles B to E.

Pause the video when you have a go at this and press Play when you're ready for question two.

And here is question two.

Here we have four quadrilaterals which are all congruent, and the transformations which map each shape onto the next one are stated below.

For example, between A and B, you can see it says A to B is a translation.

You can also see that on shape A, you've got all the angles and lengths labelled, but not on B, C, and D.

That's because in part A, you need to label the missing lengths and angles on shapes B to D.

And then in part B, you need to think about which transformation could be applied to shape D to create an image that is in the same orientation as shape A.

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 with that.

So, which transformation maps A onto B? That is a rotation.

The transformation that maps B onto C is a reflection, the transformation that maps C onto D is A translation.

D onto E is a reflection, and E onto A is a rotation.

And then once you then label on all the missing angles and all the missing lengths, you should have this.

Let's think about a few ways we can check this.

Well, you should have your lengths being three, four, and five in every triangle, and your angles being 37 degrees, 53 degrees, and 90 degrees in each of those triangles.

And in terms of where they are, your length of five should be opposite the right angle, or between the 53 degrees and 37 degrees each time.

Your four should be opposite the 53 degrees and between the right angle and the 37 degrees.

And the three is your shortest length.

That should be between the right angle and the 53 degrees, or opposite the 37 degrees, whichever way you look at it.

Then question two.

So these four quadrilaterals are congruent and we need to find the missing lengths and angles on each of the shapes B to D.

So when we apply the translation from A to B, everything stays in the same orientation, so the lengths and angles should be here on shape B.

Now, B to C is a reflection, so we reflect that over a vertical line, which means that the length and angles should be here in this case.

And then because C to D is a rotation of 180 degrees, we've got to imagine, or you can do it with some tracing paper, turning all of those measurements around 180 degrees a half turn and place 'em onto D, and we'll get 'em here.

And then finally, which transformation could be applied to shape D to create an image that is in the same orientation as Shape A? That'll be a reflection over a horizontal line.

Great job with that today.

Let's now summarise what we've learned in this lesson.

Translation, reflection, and rotation are transformations that produce an image that is congruent to the object.

That's because when two shapes are congruent, they have the same lengths and the same angles.

They might be in different orientations or different positions, that's fine, but the lengths and angles are the same.

Translation involves moving a shape to create a congruent image, rotation involves turning the shape about a defined point to create a congruent image, and reflection involves flipping a shape to create a congruent image that way.

Great job today.

Thank you very much.