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

This is our final lesson of this unit.

it's a chance for us to do some problem solving, about equations of straight lines in the context of shapes.

In particular, using lines of symmetry.

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

So pause the video and go and grab these.

And when you're ready, let's make a start.

Today's Try this task is the following.

It has a shape inside it.

There are some coordinates there.

There are some really interesting points there.

What can you say about this? And, what are the equations of the lines of symmetry in the rectangle on the right? You've got the lines of symmetry.

These are the dotted green lines there.

What can you tell me about these? What are their equations? Pause the video and have a go at this.

Okay, and now let's mark this task together.

You may have said that it has opposite sides are equal.

You may have said that it has four equal angles, each of them 90 degrees.

Okay? You probably said that this line here, labelled there as the line of symmetry, which is really good.

And you may have said that this is an interesting point.

It's actually the midpoint.

The line of symmetry cuts it in half.

So this is the midpoint.

Which means that this side here, and this side here, so these are the two parts are equal parts.

So the midpoint has cut the whole line, the whole side into two equal parts.

You will be able to say the same thing, about this point here, that this is the midpoint and we have again, the same thing.

That the side of the rectangle has been cut into two equal parts.

You may have said that, we have two lines of symmetry that intersect at 90 degrees.

We've got a 90 degree angle there.

And you would probably have said that, this is the midpoint of the other side of the rectangle.

Which means, now, that this part, and this part here are equal.

And the same thing applies obviously to the opposite side.

So we have a midpoint, and it cuts the side of the rectangle into two equal parts.

Okay.

Now the question is, what are the equations of lines, of the lines of symmetry in the rectangle? So we have our first line that we're going to look at.

We're going to look at the vertical line.

And if I want to find out, the equation of that line, first I need to know a point on that line.

Now, I know that I have the midpoint on that line.

Now to find out what are the coordinates of that midpoint, I need to look at the two points that I have there.

11, 13 and 21, 13.

I want to know the coordinates of the point, that is halfway between them.

Obviously, the y coordinate is not changing.

It's 13.

down on the same lines that will stay the same.

So I really need to focus on the x coordinate.

What number is halfway between 11 and 21? I can use a number line to help me with this.

So I can draw a number line, and have 11 and 21 on it.

And halfway between them is 16.

So now I know that, the coordinate of the midpoint here, is 16, 13.

Now on that line, on that green dotted line, any point that I choose, is always going to have x coordinate of 16.

So it will be 16, 13.

16, 12.

16, 10.

16, 7.

Meaning, it's going to always have 16 something.

So the equation of that line is x equals 16.

Now let's look at the equation, of the other line of symmetry.

So we're going to look at that horizontal one.

Okay.

If I Look at that, I need to look at these two points here.

One of the points is 21, 13.

One is 21, 7.

So we can see again, the x coordinate is the same, because both points are on the same line.

So we're looking at the y coordinate.

We want the number that is halfway between, the 13 and the seven.

We can again use a number line for this.

Halfway between seven and 13 is 10.

So now I know, that the coordinate of this midpoint here, is 21, 10.

Okay, what do I know about this line? And again, we're looking at the dotted line, the horizontal one, okay.

Any point on this line, is always going to have the y coordinate as 10.

Doesn't really matter where I choose it.

So anywhere here.

So if I take a point here, if I take it here, if I take it here, if I take it here, there will always be different x coordinates.

But, there will always be the same y coordinate.

Therefore, this line here, has the name or it has the equation y equals 10.

Okay, excellent job.

if you've done this correctly.

Let's move on to our Connect task.

For our Connect task, this is what we are going to do.

Use the table to draw shapes that match the descriptions.

And we have a table there.

We have three shapes that we want to try and draw, using the information that has been given to us.

Now isosceles triangle, we need to draw it, with a vertex location of -2, 4.

And a line of symmetry at y equals four.

Now to start with, what is an isosceles triangle? An isosceles triangle is a triangle, that has two equal sides and two equal angles.

The base angles are equal.

I would start with a sketch first, because I want to know roughly, how is that going to look like.

So I know how to draw it on the grid.

This is how an isosceles triangle will roughly look.

It will be triangle.

It has two equal sides, and it will have one line of symmetry.

And it's in the middle.

Cuts it right in the middle.

Now, the line of symmetry has to be at y equals four.

So let's go to the grid, and find out where y equal four is.

Well, the line y equal four is here.

We can label it why equal four.

Now, this is one of the criteria.

The second criteria is the vertex locations.

We know one of the vertices has to be at -2, 4.

So let's find that point.

2, 4, and that's the point.

So I know that the vertex of the triangle, or one of the vertices would have to be there.

There are different options for me here, of how to draw it.

I just need to make sure that the two sides are equal.

So I can start by saying, well, let me draw a side here.

Now what's the length of this side? Well to do that, I know that I have to go up one, and 1 2 3 4 across.

To get from one end, to the other.

So let me show you.

So I went one here, and then I went 1 2 3 4.

So I need to do the same with the other one.

But obviously, instead of going up, we're going to go down now.

So I'm going to go down one, and then across 1 2 3 4.

And I know that I'll end up somewhere here.

So I can draw it.

You can see that the two sides are equal.

You can do that on a piece of paper, by drawing the triangles actually around them.

But we should be confident enough now, that we don't need to draw these.

Okay, and now, I have the two sides, I can just add the base, connect the two points, and I have my triangle here.

Remember what I said.

We have more than one option.

I could have also done this.

Instead of going this way with a triangle, I could have gone the other way.

So I could have done a line here, a line there.

These two lines are equal and then connected them.

And I have another isosceles triangle.

And it meets the same criteria.

It has a line of symmetry at y equals four, and it has a vertex location of -2, 4.

I could have made it bigger or smaller as well.

So there's more than one possible answer.

Let's look at the second shape.

The second shape is the rhombus.

Now, what is a rhombus? Excellent.

It has four equal sides.

Really good.

Okay.

So I can start by sketching a quadrilateral.

It has four equal sides, and it has two lines of symmetry, okay? So this one has two lines, instead of one.

The criteria that has been given to us is 3, 4.

And the vertex is at 3, 4.

And the line of symmetry at x equal three.

So we'll start by drawing the line of symmetry.

So we go to the grid, find where x is equal to three and draw the line.

And label it with x equals three.

Now I need to find the vertex.

Its at coordinate 3, 4.

So I go to my grid again, go to 3, 4 and label the point.

I have the vertex and I know, we have the line of symmetry.

I roughly how the shape looks like.

So I can start by drawing the first side.

Again, it's really important to know the length of the side.

So I've gone one across, and then three down, to get to the end of this line that I drew.

So I need to do exactly the same thing, on the other side.

Go one across, and then three down, and I have my next line.

And they are equal.

Now I need to do the same thing, to get the third and the fourth lines.

Okay.

So I need to go, this time I need to, I can go one across this way to the left and then three down.

Or three down and then one across to the left.

It really doesn't matter.

And this is my side, and then this is the next one.

I can double check, because I can think, if I draw a line of symmetry across here, because we already have one line of symmetry.

If I draw another one across here.

Okay, is it giving me the mirror image? Yes, it is.

Again, more than one possible answer.

I could have made this rhombus a bit bigger, a bit smaller, a bit wider, a bit narrower.

It really doesn't matter.

You've got so many possible answers.

Provided you are making sure that each side is equal, you will end up with the correct answer.

Again, the last shape that we have here is the kite.

Okay, we've been told that the vertex location is at -4, -4, and a line of symmetry at x equal -2.

So what we are going to start, with the line of symmetry.

But before we actually get to the grid, let's just sketch it.

How does a kite look like? What are the properties of kite? Well, adjacent sides, or either side standing next to each other in a kite are equal.

Now it doesn't really matter how you draw the kite.

You can rotate it around.

So I'm going to rotate it when I sketch it.

So this is how roughly a kite looks like, okay.

And it has one line of symmetry, going through the middle, okay.

So now knowing where, that line of symmetry on the grid is, will really help us draw that kite accurately.

Go to the grid, go to x equal -2, grab a ruler, and there we go.

And we can label it as x equals -2.

Now, where is the vertex, or one of the vertices is going to be at -4, -4.

So we go there, and plot it there.

We know that this is one of the coordinates, one of the vertices.

We know that the line of symmetry is here.

Look at my sketch and look at the line on the grid.

Okay.

So on one side of the line of the symmetry, I need to draw something that looks like this.

And on the other one, I need to draw the same shape.

So my starting point should really be that line of symmetry.

If have that, I have that coordinate as one of the vertices.

So if you look at this one here, so this vertex here, okay.

It really should be this vertex, isn't it? And it doesn't, this one, Because look, you've got the line of symmetry going in the middle, line in the middle.

So we need to start from the line, and move in this direction.

So I'm going to grab a ruler, and go here and draw the first side of the kite.

Now, what have I done to do to draw this side? I went one two across and one up.

So I need to do the same, to get the other side there.

They're the one that is equal to it.

So I'm going to go one, two, and then one up.

And I know now that my line should end here.

So I can use a ruler and draw it.

Now I need to draw the other two sides.

They are equal.

They're going to meet at their line of symmetry.

So they're really easy to draw, okay.

So we can draw the first one, and just double check.

What have we done? We went 1 2 3 4 up, and then one two across.

So we need to do the same here.

1 2 3 4 up and then two across.

And that will get us there.

Or we can just join the two points, and double check that the sides are equal.

And there we go.

And we are done with the kite.

Now it is time for you to have a go at the Independent task.

Please read the two questions carefully and have a proper go at this.

Off you go.

Okay.

Now let's mark and correct the Independent task.

Looking at the shapes, shape x.

What did you write down as the name of that shape? What type of quadrilateral is that? Really good.

Excellent.

It's a kite.

We know that because adjacent sides are equal.

It has four equal sides.

Really good.

So it's a kite.

How many lines of symmetry does it have? Does a kite have? Really good.

It has one and it goes there.

And what is the equation of this line? Excellent job.

It is x equals -2 because every single point, on that line has an x coordinate of -2.

Excellent job.

Now, what about shape y? What did you write down? Really good.

It's a rectangle.

And we know that because it has, four sides to start with.

Opposite sides are parallel.

Opposite sides are equal and it has, Really good.

It has four right angles.

Okay? So it's a rectangle.

Now how many lines of symmetry does the rectangle have? Good job.

I'm glad you got two there.

Okay, so it has two.

Make sure that you do not confuse that with four, because the four is the diagonal.

And that does not give you a mirror image.

So it has the first line of symmetry going across.

What did you write down as the equation of that line? Good job.

So this line is x equals four.

And the other line of symmetry? It goes this way, and it has, Good job.

Y equal four.

Okay, shape Z.

What's the name? Really good.

It's a rhombus.

We know that because it has four equal sides.

We know that they are equal, from the triangles around the shape.

If we draw the triangles, we draw right angle triangles, we end up with the same four, right angled triangles around it.

Excellent.

So it's a rhombus.

And how many lines of symmetry does it have? Really good.

It's a rhombus.

It has one this way.

And that has an equation of x equals 10.

Well done.

And the other one? Good job.

And it has an equation of y equals two.

Really good.

Let's move on to question number two.

Draw a rectangle with vertices at -1, 2, and 3, 4.

A line of symmetry of x equal one.

So we know that one of the of the vertices, is at -1, 2.

So I can go there and plot it.

I know that the second vertex is at 3, 4.

So I can go there and plot it.

I know that it has a line of symmetry at x equal one.

So I can go to x equal one, and draw that vertical line.

I can label it and now I can start imagining.

How is that rectangle going to look like? We've got a vertex there, we've got a vertex there.

So I know I'm going to do something to connect them.

And I'm going to use that line of symmetry to help me, okay.

Now remember the line of symmetry, cuts the side of a rectangle into half.

Okay.

So here is the midpoint of that side, of that first side of the rectangle.

So if I have two units here, I should have two units there.

And this gives me my third vertex.

And now it's the same thing here.

If I have two units, I have two units from this side, and this gives me my fourth vertex.

Now all I have to do is grab a ruler and connect the vertices.

And there we go.

I have the rectangle.

If you had this correct, you should be super proud of yourself.

Well done.

This brings us to our Explore task, for today's lesson.

Draw an appropriate set of axis and complete shapes, that fit the following description.

Number one, a square with a line symmetry on x equal five.

Number two, a rectangle with a line of symmetry on y equals three, and one vertex at -5, 1.

And three, a square with a line of symmetry on x equal -4, and one vertex at -4, -1.

So this is really an opportunity for you to continue practising the skills, that we have developed earlier in today's lesson.

If not, I will give you support in 3 2 1.

So for support to start with, we need to draw the axes.

So I could start, with something that looks like this.

I have x axis, I have the Y axis and I've labelled them.

And I can look at the first description there, and then it says a square with a line of symmetry, on x equals five.

So I need to go and find where x equal five is.

I've found it and I want to draw a line.

So I draw the line of symmetry and I label it.

Now it's really important to see what have I done here.

My x axis ends at five, but that's where I also needed to draw my line of symmetry.

And I'm going to draw a square.

So I will probably need, a bit more space on the x axis.

So what I would probably need to do, is to go here.

And using a ruler, extend this a little bit more.

So you can add a six you can go up to seven.

Okay.

Now with this hint, you should be able to make a start.

Please pause the video now and have a go at this.

Now for the Explore task, these are the shapes that I managed to draw.

With the coordinates and lines of symmetry clearly labelled.

However, yours may be slightly different.

Because there are multiple solutions to this explore task.

I wonder what shapes you managed to draw.

At this point of the lesson, you should be super proud of yourself.

Because you have learned so much throughout this unit.

This brings us to the end of today's lesson.

And the end of this unit.

I'd like you to do three things for me.

The first thing, I'd like you to have a little think about the whole unit.

Everything that we've learned, and write down the three most important things from what you've learnt from this unit.

Secondly, I would like you to complete the exit quiz.

And lastly, I would know love to see your work.