Lesson video

Hello, and welcome to another lesson with me, Dr.

Saada.

In today's lesson, we will be looking at solving geometric problems. All you need for today's lesson is a pen and paper.

So please grab these, and let's begin.

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

In the diagram, one side of a square is shown.

What could the coordinates of the other vertices be? If you look at the diagram here, you have one line segment.

It shows the two vertices of the square.

If you think you can answer this question, please pause the video, and have a go at this.

If not, don't worry.

I'll give you some hints.

Okay, now, what are the properties of a square? Think about them.

Really good.

Has four equal side lengths.

It has four equal angles, each of them being 90 degrees.

What else? Opposite sides are parallel.

Really good.

Diagonals intersect at 90 degrees.

They cut each other at 90 degree.

And diagonals bisect each other.

They cut each other in half.

Excellent.

What does vertices mean? Vertices is the plural of vertex, and a vertex is the point where two or more line segments meet.

In this case, you can think about it as the corners of that square 'cause that's where the two line segments, that each side of the square meet.

It's called vertex.

Okay, now, with these hints, pause the video, and see if you can attempt the question.

Okay, let's have a look at the diagram here.

We have been given a line segment.

Now, if I try and think about that robot question that we have done in lesson two and in lesson three and think about the journey from one end point to the other end, I had to move four to the right and two up to get to this point.

Now, I can also think about this as having a right-angle triangle here.

'Kay, now, I can use this triangle to try and help me find the vertices of that square.

If I take this triangle, imagine I actually cut it and take it and put it somewhere else, like, for example, here.

Is that going to help me? The answer is no.

I cannot see where that square is going to end.

What about if I take this triangle and rotate it, just move it around a little bit? Now, before we do this, I just want to show you something really important.

Now, this here, okay, this is the first side of your square.

The square is obviously tilted.

That's the first side.

So all the sides have to be equal to this one.

Now, where is that on this right-angle triangle? It's the longest side on the right-angle triangle.

So on this one, it's actually this one here, 'kay? It's this one there.

So this needs to be the length of then my next side, okay, for the square.

Now, let's take that square, that triangle and rotate it, and maybe put it here.

'Kay, so now, that longer side is there.

Can you see? Now, I can start seeing that, ooh, I do have a 90-degree angle there.

So I have something.

I'm starting to form something here that is related to a square.

Now, let's take that triangle and move it.

We'll take it.

I've obviously marked that point as one of the vertices of the square.

Going to put it there.

Again, I'm trying to keep that longest side, so I can see where it is.

So now can you see how I'm starting to form that square? Now, I'm going to put another point here for the last vertex.

Now, if I move that triangle, if I actually put it here, it should just match.

It should show me that it's working, 'kay? And there we go.

Now, I know the coordinates of this point.

Five, three, I can read it.

I can read the coordinates of this point.

And just to double-check things, I'm going to connect them.

I'm going to connect them using lines.

There we go, and I have my square, 'kay? But hold on, that's not the only square I can form.

If I go back to that same original triangle that we looked at, so if I go to this triangle here, if I take it, 'kay, imagine I cut it.

I'm going to place it there.

I still have that longest side.

'Kay, I still have that longest side that was here.

So this side that was there, I just moved it now, and it's here.

I still have that longest side there.

I can draw it properly.

Now, if I take that triangle and move it to this side and draw the line, then I can take the triangle, ideally I don't really need it 'cause now I can see the two end points, and I can just connect them.

I can take it and just double-check that it works there.

And now I have the next possible square on the grid, with the coordinates three, negative seven and nine, negative five.

Well done if you've done this correctly.

And the solutions to the try now, without all the triangles being there on the diagram, should look like this one here.

In the diagram below, a square is enclosed by a larger square.

What are the coordinates of the missing points? Explain how you found them.

So if we look at this diagram here, we are given two points.

We want to find the coordinates of all the remaining points, and we need to be able to explain how we found them.

So let's make a start with the things that we should know.

To start with, I should know that here, this is my x-axis.

I should know that this is the y-axis.

Always start with the information that you know.

Now, I also know a very important point of this grid, and that is this point here.

And it's called the? Well done.

Origin.

And it's at zero, zero.

Now, what else do I know? We've been given this point here, which has a coordinate two, eight.

The two is the x-coordinate.

So I know now that all the points that are on this line here, all the points on this line will have an x-coordinate of two.

'Kay, so I know that here, on my x-axis, I can label this point with number two.

And therefore, I can now say that this point here has an x-coordinate with two.

I need to work out the y-coordinate now.

Well, I know that this point here has a y-coordinate of negative four.

Therefore, y at this point must be negative four.

And therefore, this point will have the coordinate of two, negative four.

What else do I know? If I look at this point here, what do I know about this point? To start with, I know that the x-coordinate is two.

I don't know about the y-coordinate yet.

But this point here is at the diagonal of the square, okay, of the inside square.

And what we said earlier is that the diagonals of a square bisect each other.

They cut each other in half.

So this is really the midpoint of that diagonal.

The diagonal has one end point at two, negative four and the other end point at two, eight.

So I need to know what's halfway between these two points.

What's the midpoint? I know the x-coordinate is two.

I need to have the y-coordinate.

So I want to know what's halfway between negative four and eight.

I can use a number line to help me with this.

So I can quickly sketch a number line.

Negative four to eight, that is 12 places.

I want half of that.

So I want to go from negative four up six places.

From negative four, if we add six or go up six places on the number line, we end up at? Excellent.

We end up at two.

So now I know that on my y-axis, this point here is two.

And therefore, that coordinate of that point is two, two.

'Kay, now, if we look at this point here, it tells us that the x-coordinate at this point is eight.

So I can go to the x-axis and label this point as eight.

Now, I have two more points on this line.

I can say that this point is eight, something and that this point is also eight, something.

I need to work out the y-coordinate of each of them.

Well, for the first one, it's quite easy because we have already worked out that the y value on that line there is two.

So I can say that this point is eight, two.

All the points on this line will have a y-coordinate of two.

The one at the top there will have a y-coordinate of eight.

Because I can see, from here, I can see that this point had a y-coordinate of eight.

So all the points on this line are going to have a y-coordinate of eight.

'Kay, now I have three more points left.

Let's look at the base of this square here.

To get from this point to this point, I have to move six places to the left, to move from eight to two.

Now, if this point is the midpoint, I've moved six places to the left.

So to get to the end, I need to move another six places to the left.

From two, if I move six places to the left, that gets me to negative four.

And we already know that the y-coordinate at this point is going to be negative four.

We've already established that here, 'kay, when we looked at the y-coordinate.

Now, for the next point, we know that the x-coordinate here now is negative four.

So this point here, so if we know that the x-coordinate for this point is negative four, then the x-coordinate for this point will also be negative four.

And we know that the y-coordinate is two.

And for the last point, again, we know that the x-coordinate is going to be negative four, and we already know that the y-coordinate for that line, for any point on that line, is eight.

And now we have worked out the coordinates of every single point that was missing, and we managed to explain how we work them out.

And now it is your turn to independently practise some of the skills that we've learned today.

Please pause the video, and have a go at questions one and two.

If you need help, hold on.

I'll give you some support.

And for support, let's read the question together.

Find the coordinates of the remaining vertex for each of the squares.

So for part a, you've been given three points that are three vertices of a square.

You need to find the fourth one.

Think about what's happening from one point to the other.

You can start with any of them, any of the points.

It really doesn't matter.

I'm going to start with one, negative two.

From one, negative two, to get to one, two, I'm going up four places.

So I'm going up four.

That's the distance I'm covering.

What am I doing here, from this point to negative three, two? Think about that.

Think about the properties of a square.

The sides must be? Yeah, you're right.

They must be equal.

So I should be going how many places? Yeah, Four, four spaces.

And what do I need to do here? Think about this.

Going to give you a hint for part b as well.

Part b does not have a number.

It doesn't have any coordinates.

But what we can do is to do this.

Think about it as a journey.

Remember that robot question that we did in lesson two.

If it's a robot moving from one point to the other, the robot will have to go one down and then three to the right.

You need to do the same to get from one point to the other.

Not necessarily every time one down, but you need to be doing one and a three to make a triangle.

'Kay, use those two hints to help you make a start.

Now, pause the video, and have a go.

Now, let's mark and correct the independent task.

For part a, if you look at the points that have been given to you, I have point one, negative two.

And if I think about the journey from that to point one, two, what have I done? I went up four.

From one, two, to get to negative three, two, what did I do? Really good.

I went to the left four.

Now, it's really important to notice that the distance between each vertex from one vertex to the other is four.

And that is because the sides of a square are equal, so the gap or the distance between each vertex is going to always be the same.

So now, if I want to find where the fourth vertex is, from one and negative two, I need to move four to the left.

And I can also double-check by going now up four spaces.

If I get to negative three, two, then it's correct.

Now, I want to find the coordinate of the vertex.

So here, I have the x-coordinate is one.

And if I move four to the left from one, that gets me to? Excellent job.

Negative three.

And y-coordinate is staying the same.

All the points on this line are always going to be negative two because y here is negative two.

So the coordinate of the point is negative three, negative two.

Well done.

Okay, and next one, you have been given three points.

This time, I'm not giving you the x- and the y-coordinates.

I want you to find the last point or the last vertex of the square.

So I've asked you to look at the points and think about either the robot question that we have done in lesson two and three, or think about right-angle triangles.

So let's think about right-angle triangles.

Think about from this point to this point, 'kay? To get from here to there, I need to go one down and three to the right.

Where to connect them, can we see that right-angle triangle? Really good.

So this here, the purple side, is the side of the square.

Now, the next point is here, so I know that the next side's going to go somewhere there.

Now, what should the distance from this point to this point be? Should be exactly the same.

Should be one and a three.

Now, this time, we're going to go one to the right and three up, and I'm going to join them.

And now I can see here, the square is starting to form.

I can see that the square, in this case, is tilted, 'kay? Now, I want to know the last point or the last vertex of the square.

I know it's going to be somewhere here.

But where exactly? Is it here? Is it there? Is it there? 'Cause all the four sides have to be equal.

I can carry on doing the same thing with the one and three from this point or from here.

It really does not make a difference.

I'm going to do it from here.

So I know it's somewhere here.

I going to go up three and one to the right.

So up three and one to the right, that gets me to this point.

So this is that remaining vertex of the square.

Now, I cannot say what the coordinate here is exactly because I don't know the coordinates for any of the points to work this out.

But I can mark it clearly.

Now, I'm going to join them, and you can see now you've got three sides of the square.

So we can see that we've got right angles here.

Now, to double-check our answer, from this point to this point, we should also have another one and three.

So if I join them on double-check, I go across three and one down.

So this tells me that this side is equal to this side, is equal to this side, is equal to this side.

Well, this line segment is equal to this line segment, is equal to this line segment, is equal to this line segment because the triangles or the right-angle triangles formed around them are exactly the same.

Well done if you had this correct.

Question two, find the coordinates of the remaining vertex for each of the squares described.

Three of the vertices are one, one, three, three, and three, one.

Now, some of you are going to be able to look at this question and tell me the coordinate of the remaining vertex of the square.

Some of you would prefer to do a sketch first.

I certainly prefer to sketch this first, okay? 'Cause it helps me see the question, see what the square looks like, see where my solution is, and justify my answer.

'Kay, to start with, I'm going to start by drawing x- and y-axes.

Notice that it's a sketch.

It's a really quick sketch.

It's not accurate.

I'm going to write or label my one and three on both x- and the y-axes.

Just because if I look at the numbers in my question, I'm only given one and three in part a.

Now, the first point that I've been given is one, one, so I'm going to say it's roughly there.

The second point I'm given is three, three, which is roughly here.

The third point is three, one, and that is roughly here.

'Kay, now I'm going to consider the journey from one point, one vertex to the other vertex of the square.

I want to think about the distance between the vertices here of the square.

If I go to the first point, one, one, and think about how do I get from that to three, one, I move to the right two places.

Now, if I want to move from three, one to three, three, I move up two.

Notice, in both cases, I did a movement of two.

And that is because the square has four equal sides, so all the sides are going to be equal.

If I move two once, I have to move from the other vertex two and to the other vertex two and so on.

So now I can think, okay, what do I need to do next? Well, I definitely need to go up two from here to get to that last vertex of the square.

What would be the coordinates of this one? It will be? Excellent.

One, three.

So I moved from one two places up, so that gets me to three on the y-axis, whereas the x-coordinate stays at one.

So that answer for the first part of question two is one, three.

Really good job if you've done this correctly.

Question two, I need a bit more space to show you the answer, so I'm just going to go to the next slide.

Okay, and for the second part, you were told that two of the vertices are negative two, four and zero, two.

So I'm going to start by doing exactly the same thing.

I'm going to start with sketching x- and y-axes, and I'm going to label the points or sketch where the points roughly are going to be.

So zero, two will roughly be there.

And negative two, four, I know it's going to be in the second quadrant, so it will be somewhere there.

Now, I'm going to connect the two using a line.

So I know that this is the first side of my square.

So I can start to visualise, where is that square going to be? 'Kay, and I'm going to think about that robot question that we did earlier in lesson two and lesson three.

I'm going to think about the journey from negative two, four to zero, two.

So either think about the robot, or think about drawing right-angled triangles.

'Kay, so to get from that point, I need to go down, and then I need to go to the right.

I'm going to go down two and to the right two.

Now, my next point, 'kay, my next vertex is going to be somewhere here.

I know it's going to be somewhat here because it looks like I have a tilted square that I'm going to draw like this, 'kay? So if I want it like that, somewhere around this area, I need to go across two and then up two.

'Kay, and that's what we are going to do.

So I'm going to go two to the right and then two up, and that gets me to this point.

It has the coordinate two, four.

And I know that that coordinate is two, four because here I had x is equal to zero.

If I go two across the x-axis, I get two, x-coordinate being two.

And I had the y-coordinate of two.

And I went, at this point here, and I went up two places, so I'm going to get a y-coordinate of four.

Really good.

Now, I'm going to connect those two points using a line.

So I can start to see, how is that square forming? Now, I can do the next point from here or from there.

It really doesn't matter.

I'm going to go and do it from negative two, four.

So I'm going two, this time, I know that the point is going to roughly be here.

So I know I'm going up, and then I'm going to the right.

'Kay, so I am going to go up two and to the right two, and that gives me my last point.

And again, I know the coordinate here.

Because at this point, the x-coordinate was negative two, and I went two places from negative two.

That gets me to zero.

'Kay, that gets me to x-coordinate of zero.

And I know that the y-coordinate here was four.

And I went up two, so that gets me to a y-coordinate of six.

I'm now going to use a line and connect the points.

And finally, I'm going to connect it this way.

Now, to check my answer, I should be able to go two and then two down, and that should get me to this point.

And now I have the points for the last coordinates or coordinates of the other two vertices.

So these are the two that were given to me, and zero, six and two, four are my solutions.

Now, remember, you could have worked out a slightly different square.

I went this way up.

You could have gone this way down and then another square here, like this way, going this way.

Okay? It'd be really interesting to see what you have done.

And you've done some amazing learning so far, and now this is a really, really interesting task.

We've gotten to our Explore task of today's lesson.

Let's read it together.

The large square encloses that pale grey square in the diagram below.

Find all the missing coordinates.

Now, we've done a question similar to this at the beginning of the lesson.

But this one, I tried to make it a little more challenging and actually more interesting for you.

Try and remember when doing this question that the square has four equal sides.

Try and think about the fact that you've got diagonals that bisect each other.

They cut each other in half, which means you have to think about midpoint.

Now, if you are confident enough to make a start on this question, please pause the video now, and have a go.

If not, don't worry.

I'll give you some hints.

I'm going to suggest that you start by looking at this point here.

It has the coordinate two, four because it's the midpoint of that line segment.

Halfway between one and three is two, and halfway between one and seven is four.

Now, my second hint for you is to think about right-angled triangles, how to get from one point to the other.

Think about this.

Now, pause the video, and have a go.

I wonder how you got on with this question.

These are the solutions, 'kay? These are all the coordinates for all the points.

Please mark and correct your work.

And this brings us to the end of today's lesson.

You've done some fantastic learning today.

You should be really, really proud of yourself.

Please remember to complete the exit quiz.

And enjoy the rest of your day.

Bye!.