Lesson video

Hello and welcome to this lesson on quadrilaterals in circles with me, Miss Oreyomi.

For today's lesson, you've be needing a pen, a pencil, a ruler, and also your book.

So if you need to pause the video now to go get your equipment, then please do so.

Also try to minimise distractions by putting your phone on silent and also trying to get into a space with less noise.

So do so now, and resume when you're ready to start the lesson.

For your try this task, how many different quadrilaterals can you draw using the dots where this diameter is your diagonal? So how many quadrilaterals can you draw where this diameter is your diagonal of your shape? So you can either trace the circle again and the dots onto your sheet of paper, or you can get a rounded shape, trace that shape on your paper, and then put eight dots around the circle and draw the diagonal from here to here.

Now pause the screen and attempt to draw as many different quadrilaterals as you can using that diameter as a diagonal of your shape.

Okay, hopefully you had a go at that.

These are the several examples you could have drawn using that diameter as a diagonal of your shape.

So you could have drawn a kite, square, a scalene quadrilateral and a rectangle.

I wonder if you've got some of those as well.

Okay, two students are having a discussion about using their knowledge of circles to describe quadrilaterals.

And they're asking based on our knowledge of quadrilaterals, can I determine the quadrilateral that is shown below? Student A is saying, "I know the sides are the same length because dot dot dot." Can you help student A explain why the sides are the same length? Hopefully you're thinking along the lines of, well, there is one dot separating each connected one.

So there's one dot here, one dot here, one dot here, and one dot here.

Therefore, all the lines must be the same.

If we also think about the radius.

Well, from the centre of my circle to the end of my circumference is the same length as going from here to here, going from here to here, going from here to there and going all the way around my circle.

So therefore they must all be the same length.

So that is one way that we can use our knowledge of circles because the radius is the same all around my circle, this length must also be the same, the sides must also have the same length.

Okay, let's go to point number two then.

"I know the diagonals are perpendicular because." How can you help this student explain that he knows that the diagonals are perpendicular? Well, firstly, how do we figure out the diagonals of a quadrilateral? If you said we connect up the vertices, then you are absolutely correct.

So I have connected up the vertices of my diagonals.

What does perpendicular mean? Essentially when lines cut each other at right angle, isn't it? So my diagonals are perpendicular because they cut each other at right angle.

So we've been able to explain, using our knowledge of circles to explain that this shape, this quadrilateral, is therefore a rhombus because all the sides are equal and it has two lines of symmetry.

Also the diagonals are perpendicular to each other.

Okay, let's discuss this statement that Xavier's made.

He says he has drawn a kite.

Is he correct? If you think he's correct or if you don't think he's correct, how do you know? What features of the shape can you describe? Take a moment or two now to just think about this and then when you're ready, resume the video and we can discuss the features.

Okay, he says he has drawn a kite.

Well, what do we know about a kite essentially, about the properties of a kite? Well, first property I'm going to go for is that there's one line of symmetry here.

Okay, so there's one line of symmetry.

Also, if I measure my radius from the centre of my circle to one point of my circumference, it is one square.

So it's the same radius from here to here, that is also the same length here.

And it's also the same length here.

So I can say that this length and this length are the same.

So I know for a kite that adjacent sides have the same length.

What of this side? I've got one, two, three dots between my connected dots.

And I've got one, two, three dots between my connected dots here.

So again, I could say this side is also the same as this side.

So yes, I do agree that Xavier has drawn a kite because adjacent sides are equal and also I have one line of symmetry.

Your turn then.

Xavier formed a quadrilateral in a circle.

Can you form quadrilaterals in a circle by connecting dots on both circles? So this circle here and this circle here.

Can you create a quadrilateral that has two lines of symmetry for the first one here? So for this one, you're creating a quadrilateral that has two lines of symmetry.

And for this one, you're creating a quadrilateral with one line of symmetry.

If you need to get an object that is round for you to trace over and put the dots around the circle, then please do so.

Or you can trace, you can put your paper and trace over the screen and draw as many quadrilaterals as you can that has two lines of symmetry by connecting the dots.

And also that has one line of symmetry by connecting the dots.

So pause your video now and attempt this task.

When you're done, resume the video, and we can check the quadrilaterals that you drew.

Okay, here are some of the ones I came up with.

So for the first one, it is a rhombus.

It's got two lines of symmetry, one here and the other here.

This is a rectangle.

I've got my delta here and I have my kite over here.

Did you manage to get the same ones or did you get different ones? It is now time for your independent task.

So pause the video now and attempt all the questions on your worksheet.

Once you've done that, come back and we'll go through the answers together.

Okay, how did you get on with that? I hope you were able to answer as many questions as you possibly could have.

Let's look at A then.

So we want to work out what the value of A is.

This is two centimetre.

Can you see that this two centimetre, if I make it into a straight line, is the same as A? So A is going to be two centimetres.

I'm just going to write this here as well, so that you can see that A is two centimetre.

If I join this two centimetre and then I join another one here.

Let me just change the colour for one second.

So if I put this here as well, do you see how this is going to be two centimetre? Now, if I take both of this and I transfer it here, do you see how the diameter of this circle is the same as this length B centimetre? So B is four centimetre.

If I change the colour again for C, well, C is two and a little bit extra, isn't it? So it's the radius A and the little bit extra.

So C is three centimetre.

So check in there you've gotten that right.

Next one.

Find the side lengths of each quadrilateral drawn on the centimetre square grid.

I've written the answer on the screen for you.

So check in your work.

Okay, number three.

Imagine moving the black vertex, so this point here, of this quadrilateral to each of the labelled points.

So if I move this point here, if I move it here, what shape are we going to have? Well, A, we're going to have a delta, aren't we? If I move this point to B, so imagine we have this, this, this and that, B is going to be a kite.

And if I move.

I'm just going to rub things out.

If I move the point from here to here, so if I connect the point up we would have a rhombus.

So A.

Use a different colour very quickly.

A is a Delta.

B is a kite.

Keep doing that.

B is a kite right here, and C is a rhombus over there.

Now each image is created using circles with a radius of two centimetre, four centimetre and six centimetre.

Name the following shape, find their side lengths and describe their symmetry.

Well, we know that A is a delta, and if this is two centimetre, two centimetre.

So this length here is four centimetre.

And we have two, two, two.

So this length here is six centimetre.

And because it's a delta and it's got one line of symmetry, the side lengths are reflected.

And the rotational symmetry is one as well.

This is a parallelogram.

Okay.

Well, this length here is going to be two centimetre, and this is going to be two, four, six.

So this is going to be six centimetre.

Our order of rotational symmetry is two and the parallelogram has got zero lines of symmetry.

So that is zero.

Number three, we have a kite.

This is four centimetre right here.

And this length is two, four, six.

This is six centimetre.

Our order of rotational symmetry is one.

And our line of symmetry is one as well.

Last one, we have a rhombus here.

All our side lengths are the same in a rhombus so it's two, four, six.

So each side length is six centimetre.

Then we have our two lines of symmetry.

And our order of rotational symmetry is two as well.

So check in your work and comparing that with the answer on your screen.

Let's move on then to our explore task.

The four identical circles, so circle one, two, three, four, with a radius of three centimetre, they intersect at their circumference and centres.

So they intersect at the circumferences and at the centres.

What shape is shown below? Can you describe the features of the shape you can see on your screen? So I have given you some points to think about, think about the side lengths, think about the rotational symmetry, think about the interior angles.

I'm really interested to see how you go about working this out.

What are the interior angles for this shape? And think about the diagonals as well.

Pause your screen now.

Attempt this.

Once you think you've described as many features as you can about the shape, resume your video, and we'll go over some of its features.

Okay.

The shape was a parallelogram.

Two of the sides is six centimetre.

So we've got three centimetre, three centimetre.

So this would be six centimetre.

Here we have three centimetre, three centimetre, six centimetre here.

The order of rotational symmetry is two.

I am going to say, I'm just going to tell you, that this angle is 120 degrees and this angle is 60 degrees.

That's all I'm saying.

If you've managed to prove why that would be excellent.

So if you want to go away from this task to prove why that angle would be 120 degrees and why this would be 60 degrees, then by all means do so.

And then we have diagonals that bisect each other.

I'm going to use a different colour pen to show this, diagonals that bisect each other.

Do you remember what it means when we say bisect? Diagonals that cut each other in half.

Right.

We have reached the end of today's lesson.

I quite had a lot fun delivering this lesson and working through the exercises, and I hope you did too.

Do not forget to complete the quiz, just to consolidate your knowledge and to help you know what you've learned from today's lesson.

And I will see you at the next lesson.