Lesson video

Hello, my name is Dr.

Rowlandson and we have a great lesson in store for today, so let's get started.

Welcome to today's lesson from the unit of circle theorems. This lesson is called the angle at the centre of the circle is twice the angle at any point on the circumference.

Now this is what is known as a circle theorem, and by the end of today's lesson we'll be able to derive that theorem and also use it.

Here are some previous keywords that will be useful during today's lesson, so you may want to pause the video if you want to remind yourself what any of these words mean and then press play when you're ready to continue.

The lesson is broken into two learning cycles.

In the first learning cycle, we're going to be investigating angles using circular geoboards.

This is so we may observe specific cases where this circle theorem occurs.

And then, in the second half lesson, we'll introduce the circle theorem more formally, look at how to derive it and how to use it.

But let's start off with investigated angles with circular geoboards.

A geoboard is a mathematical tool to explore aspects of geometry.

It's often a board containing pegs laid out in a pattern.

And patterns can vary between boards.

For example, here we have three different geoboards.

Polygons can be made on a geoboard by wrapping elastic bands around the pegs.

And we can see examples of three triangles that have been made on these geoboards.

Here we have Jacob.

Jacob is using a circular geoboard that contains 10 points equally spaced around a circle and a point in the centre.

He makes a triangle.

He says, "I wonder what size the interior angles are in my triangle." How might Jacob work out the size of each of these interior angles? Now we're going to work through this together shortly, but perhaps pause the video while you think about this yourself and then press play when you're ready to continue.

Let's see what Jacob does.

He says, "The points around the circle are equally spaced.

So the angles at the centre of each sector are equal." Now the angles around a point some to 360 degrees and there are 10 sectors.

So we can do 360 divided by 10 and that'll give 36 degrees for each of those angles in the centre.

We're concerned with the angle that is in our triangle.

So we know this angle here, this one is 36 degrees.

Jacob says, "I can use other properties of the circle to work out the size of the other angles in the triangle." In particular, we can use the property that the radius is the same length in every direction on a circle.

That means two sides of this triangle are radii, so they are equal in length.

And that means that the triangle is isosceles, so the two remaining angles are equal.

Now interior angles in a triangle sum to 180 degrees.

So to work out the two remaining angles, we could do 180, subtract 36 and that'll give 144.

So the two remaining angles must sum to 144 degrees.

They're both equal, so we can do 144 divided by two to get 72 degrees per angle.

And Jacob justifies this by saying, "I divided by two as the base angles in an isosceles triangle are equal.

So let's check what we've learned.

Here we have three circular geoboards.

You've got a triangle in each one.

Calculate the interior angles in these triangles.

Pause while you do this and press play when you're ready to see some answers.

Here are your answers, check these against your own.

With the one on the left because it's got five points around the circumference, you could start by dividing 360 by five to get 72 degrees, then subtract from 180 and divide by two to get the two 54 degree angles.

So let's go back to Jacob and his circular geoboard that contains 10 points equally spaced around the circle and a point in the centre.

He makes another triangle and says, "I wonder what size the interior angles are in this triangle." Now this triangle is different to the last one we saw in that this time Jacob has not use two consecutive points on the circle.

So how might Jacob work out the size of each interior angle this time? Consider what parts of our work it would be similar to last time and what parts of our work would need to be a little bit different.

Pause the video while you think about this and press play when you are ready to continue.

Let's take a look at this together.

We could imagine this circular geoboard split up into its 10 equal size sectors again, and we know that the angle at the centre of one of these sectors can be worked out by doing 360 divided by 10 to get 36 degrees.

Now the angle in Jacob's triangle takes up three of these sectors.

So that means the angle that he formed at the centre of the circle is equal to three lots of 36 degrees, therefore it must be 36 multiplied by three, which is 108 degrees.

Now in this case we've done this calculation as two separate steps.

We divided 360 by 10 and then we multiplied our answer by three, but we could consider this in terms of fractions.

There are 360 degrees in a full turn.

Each sector of this circle is one 10th of that.

And we have three sectors that are covered by Jacob's angle, so that means his angle is three tenths of 360 degrees and that would give 108 degrees as well.

From here, the rest of our working can be done in the same way.

Jacob says, "This triangle is also isosceles as the two sides are the radii of the circle.

So I can use this fact to find the other two angles." We can subtract 108 from 180 to get 72 for the two remaining angles, and divide it by two because those two angles are equal, that will give 36 degrees.

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

Here we've got three circular geoboards and you've got a triangle drawn in each of them.

Could you please calculate the interior angles of each triangle? Pause while you do it and press play when you're ready for answers.

Let's take a look at some answers.

Each triangle isosceles because two of the sides are the radii of the circle.

So that means if we can work out the angle at the centre of the circle, which is in each triangle, we can then work out the other two remaining angles which are equal.

For the one on the left, we can work out the angle at the centre by doing two ninths multiplied by 360.

The ninths in the denominator, that relates to the fact that there are nine pegs on this circular geoboard or other words, the circle is split into nine small equally sized sectors.

The two in the numerator relates to the fact that the angle at the centre of the circle, which is in the triangle, takes up two of those sectors.

Once we've got that and we've got 80 degrees for one angle, we can subtract from 180 degrees and divide by two to get the other two angles of 50 degrees.

So let's go back to Jacob again with his 10 point circular geoboard.

And so far with all the examples we've seen, one of the vertices of the triangle has been at the point in the centre of the circle.

And Jacob says, "When one of the vertices of my triangle is at the centre of the circle, the triangle is isosceles." This is because two of the sides of the triangle are radii of the circle.

But maybe Jacob doesn't want to make a triangle where one of its vertices are at the centre.

Maybe he wants to make a triangle.

It looks a bit more like this.

He says, "This time, I've drawn a triangle where all three vertices are at the circumference of the circle.

Is this triangle still isosceles?" Hmm.

Well, there are a few different ways we can reason about this, but one way we could do it is to find the interior angles of the triangle.

So how can we go about doing that? Well, Jacob says, "I can draw in the radii of the circle to create three isosceles triangles within this triangle like so." And then he says, "I can work out the interior angle of these triangles." And we can do that using the same methods we've done so far.

For example, for the triangle at the top we can work out the angle at the centre of the circle by doing three tenths of 360 degrees.

That gives 108 degrees.

Subtracted it from 180 divide by two to get the other two angles of 36 degrees.

The one on the right would have the same angles.

The angle at the centre is again three tenths of 360 degrees, so the other two are 36 degrees.

And for the triangle on the left, well, the angle at the centre this time would be four tenths of 360 degrees, which is 144 degrees.

So which means the other two angles are 18 degrees.

Now we don't actually have yet any of the angles for Jacob's actual triangle, but what we can see is that each of the interior angles of Jacob's triangle is split into two parts by the radii that intersects it.

That means each of those angles is a sum of its pair of angles at that point.

In other words, to work out each angle in Jacob's triangle, we need to add together the two angles at those points, 54 degrees, 72 degrees and 54 degrees.

That means yes, this triangle is isosceles as two of the interior angles are equal.

But will this always be the case? We've already seen that when one of the vertices of the triangle is at the centre of the circle, it will be isosceles because two of its sides will be the radii of the circle.

But will it still always be the case when all three vertices of a triangle are on the circumference of the circle? Perhaps pause the video while you think about that and press play when you're ready to continue.

Let's check this for yourself now.

Here we have a circle geoboard that contains nine points equally spaced around circumference, and you've got a triangle where all three of its vertices are on the circle.

Could you please calculate the interior angles of this triangle to determine whether or not it's isosceles.

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

Okay, let's take a look at some answers.

If you split this triangle into three triangles by drawing in the three radii from the centre of the circle to each of its vertices, you can work out the three angles at the centre by doing a fraction of 360 and then work out the angles at the circumference for each triangle.

And then, you would need to add together each pair of angles at the circumference to work out the angles for the actual triangle.

That would give you 80 degrees, 60 degrees and 40 degrees.

Now we can see here that there are no two angles that are equal to each other.

Therefore this triangle is not isosceles.

All the interior angles are different sizes.

Let's go back to Jacob who now compares two of the triangles that he previously drew on the 10 point geoboard and to make it easier to compare, he labels some points.

If we look at these two triangles, what we can see is that they have different angles, but these two triangles do have something in common.

The chord AB is a side of both of these triangles, and that chord AB is the same in each circle.

On each circle geoboard, that chord starts in the same place and ends in the same place.

Therefore, the main thing that differs between these two triangles is the position of the angle which is subtended by that chord AB.

In other words, the angle which is opposite the chord AB.

With the triangle on the left, that angle is at the centre of the circle, whereas on the triangle on the right, that angle is at the circumference.

Now Jacob notices that there is a multiplicative relationship between the angles that opposite the chord AB, in each triangle.

With the triangle on the left, that angle is 108 degrees, whereas with the triangle on the right, that angle is 54 degrees.

Can you spot what Jacob has spotted? He says, "Angle AOB is subtended by the chord AB, and this angle is at the centre of the circle." That's the one we can see on the left.

"Angle ACB is also subtended by the chord AB, and this is at the circumference of the circle." That's what we can see on the right.

He says, "The subtended angle AOB is 108 degrees, and this is two times the size of subtended angle ACB." In other words, both of those angles are opposite the chord AB, but the one in the centre of the circle is twice the size the one at the circumference of the circle.

Now, when mathematicians spot relationships or patterns within numbers, it can be quite exciting because it can make you wonder whether or not that relationship is always true for all cases, or whether it just happens to be true in this particular case.

You can just sometimes have one number being double another number by pure chance.

So Jacob wants to investigate if this multiplicative relationship works for other triangles drawn on the circular geoboard.

Let's take a look at what he's got.

He says, "The chord BC is a side of both of the triangles, so I'm going to compare the subtended angles." Now on the one on the left, angle BOC is subtended by the chord BC, and this angle is at the centre of the circle, that's 144 degrees.

And with the triangle on the right, the angle BAC is also subtended by the chord BC, and this angle is at the circumference of the circle, that's the 72 degree angle one.

The angle BOC is 144 degrees and this is double 72 degrees, which is the size of the angle BAC.

So yes, once again we've seen another example where the angle at the centre of the circle is double or twice the angle at the circumference where both those angles are subtended by the same cord.

Let's take a look at another example.

Here's a third one.

Can we see the same situation happening here? Jacob says, "I notice for the angles subtended by the same cord, AC, that the angle at the centre is double the size of the angle at the circumference." We can see that for the triangle on the right, the angle which is opposite the cord AC, or is subtended by the cord AC, that one's at the circumference and it's 54 degrees.

And if we multiply it by two, we get 108 degrees, which is the angle at the centre, on the left.

Okay, so now for Task A.

This task contains four questions and here is question one.

You've got three circular geoboards and you need to calculate the interior angles in each triangle.

Pause while you do it and press play for more questions.

Here is question two where you have three more triangles to find the interior angles off.

Pause while you do this and press play for question three.

Here is question three, pause the video while you do this and press play when you're ready for question four.

And finally here is question four.

Pause while you do this and press play when you're ready for answers.

Okay, let's take a look at some answers.

Here are the answers to question one.

Pause while you check these against your own and press play for more answers.

Here are the answers to question two.

Pause while you check these against your own and press play for more answers.

Let's now take a look at question three, it's a little bit more complex.

And then you need to write down what you noticed about angles POR and PQR.

Explain why you think this is.

Well, angle POR is equal to two times angle PQR.

In other words, the angle at the centre is twice the angle at the circumference.

And with question four, you had to work out the size of these unknowns.

A is 60, B is 60, C is also 60, D is also 60, but E is a little bit different.

You work out E, you should get one 120.

And then you have to write down what you notice about your answers.

Well, in each case the angle is subtended by the same chord.

But what you might also notice is that angles A, B, C, and D are equal, but E is not.

You're doing great so far.

Now let's move on to the second part of this lesson where we're going to introduce a circle theorem more formally.

Here we have Jacob who has noticed that for angle subtended by the same cord, the angle at the centre is twice the angle at the circumference.

And we can see an example of that here on this circular geoboard.

Jacob says, "Does this only work for drawing angles on circular geoboards or will it work for any circle?" Well, one way that we can explore lots and lots and lots of different examples of a phenomenon is by using dynamic geometry software.

And if you have access to this slide at the bottom of it, there is a link to a geo algebra file that allows you to investigate this further yourself.

It looks something a bit like this.

In the examples we're about to look at, we are particularly focused on the relationship between the angle at the centre of the circle, angle AOC, and the angle at the circumference of the circle, that is ABC.

Both of those angles are subtended by the same arc, AC, or the same chord, which goes from A to C.

We can't see the chord though on this diagram here.

Let's take a look at what happens as we change the angles.

The angle at the centre is now 140 degrees and the angle at the circumference is now 70 degrees and we can see that yes, the angle at the centre is still twice the angle at the circumference.

And let's look what happens as we move the position of point B.

The angle at the centre is still twice the angle at the circumference.

They are both still 140 degrees and 70 degrees.

So Jacob says the angle AOC is always twice the size of the angle ABC, no matter where I move point B on the circumference, except when I move point B beyond point A or C, the angle ABC changes like this.

Why is this? Hmm.

Well, whilst the marked angles are both subtended by the same cord, AC, they are not both in the same segment.

One is in the major segment, that's the 140 degree angle, whereas the 110 degree angle is in the minor segment.

We could also say that the angle 140 degrees is subtended by the minor arc that goes from A to C, whereas the angle which is 110 degrees is subtended by the major arc that goes from A to C.

So while they are subtended by the same chord, AC, they're not subtended by the same arc.

Jacob says, "For the relationship to work, the angle at the centre and the angle at circumference need to be subtended by the same chord and the same arc." Angle ABC, which is the angle at circumference, is subtended by the major arc AC, which is this one here.

Now the angle at the centre subtended by that same major arc is in fact the reflex angle AOC.

So it's not the 140 degree angle, it's the larger angle, which is the reflex one.

Jacob says, "I know that angles around a point sum to 360 degrees, so the reflex angle AOC must be equal to 220 degrees.

And 220 degrees is double the size of the angle of ABC, so yes, the relationship still works." So when the angles are subtended by the same arc like this, the angle at the centre, which in this case is AOC, is twice the size, the angle at the circumference, which in this case is ABC, we can see we've got 120 degrees at the centre and 60 degrees at the circumference.

Jacob says, "No matter where I move points A, B, or C on the circumference, there'll always be this relationship." We can see it again here.

The angle at the centre, 151 degrees, is twice the angle at circumference, which is 75.

5 degrees.

You can see it here.

The angle at the centre, 178 degrees, is twice the angle at circumference, which is 89 degrees.

We can see it here as well, even when the line segments cross.

The angle at the centre is 35 degrees and the angle at circumference is 17.

5 degrees.

The one at the centre is twice the one at circumference.

But here we have a situation where it looks like it's not the case, but Jacob says, "If this isn't the case, I need to check that the angle at the centre and the angle at circumference are in fact subtended by the same arc." In this case, they're not, the angle at circumference is subtended by the major arc, whereas the angle at the centre is subtended by the minor arc.

So in this situation, if we work out the reflex angle at the centre, which is 216 degrees, we can see that it is still double the angle of circumference, which is 108 degrees.

So let's check what we've learned.

True or false, the angle at the centre must always be an obtuse angle.

Is that true or is it false? Pause while you make a choice and press play for an answer.

The answer is false.

So why is it false? Pause the video while you write down a sentence and press play when you're ready for an answer.

An example of an answer for why it's false could be that the angle at the centre of a circle could be any angle between zero and 360 degrees, depends on where the points are on the circumference.

For example, this one is an acute angle and this one is a reflex angle.

So here's another true or false question.

If angle DOF is 130 degrees, then angle DEF is 65 degrees.

Is that true or is it false? Pause the video while you choose and press play when you're ready for an answer.

The answer is true.

So why is it true? Pause the video while you write down a sentence or two to explain and then press play for an example answer.

So why is it true? The angle at the centre of a circle is twice the angle at circumference if the angles are subtended by the same arc, which in this case we can see they are both subtended by the minor arc DF, which is now highlighted on the screen.

Here's another true or false question.

If two angles are subtended by the same chord, then they are also subtended by the same arc, and as a diagram on the screen to illustrate it.

Is that true or is it false? Pause the video while you choose and press play for an answer.

The answer is false.

So why is it false? Pause the video while you write down a sentence or two and press play for an answer.

So why is it false? Well, in the diagram, we can see that both angles DOF and DEF are in fact subtended by the same chord DF.

However, DOF is subtended by the minor arc DF, and angle DEF is subtended by the major arc DF.

Another true or false question, if angle DOF in this diagram is 130 degrees, then angle DEF must be 65 degrees.

Is that true or is it false? Pause the video while you choose and press play for an answer.

The answer is false.

So why is it false? Pause the video while you write down a sentence or two to explain and press play for an answer.

Why is it false? Well, the angle at the centre of a circle is twice the angle at the circumference, only if the angles are subtended by the same arc.

Angle DOF is subtended by the minor arc and angle.

DEF is subtended by the major arc DF.

So no, one is not twice the other.

So let's work out what the angle really is.

If angle DOF is 130 degrees, calculate the size of angle DEF.

Pause the video while you do that and press play when you are ready for an answer.

Okay, let's take a look.

The angle DOF is 130 degrees, but the reflex angle DOF, which is the one at the centre, and the angle DEF, at the circumference, are subtended by the major arc DF, so what we want is a reflex angle.

We can do 360 degrees, subtract 130 degrees to get the reflex angle of 230 degrees.

And then we can use the fact that the angle at the centre is twice the angle at the circumference by dividing by two to get 115 degrees.

So we have looked at lots of examples that have formed a theory that the angle at the centre is twice the angle at any point on the circumference if the angles are subtended by the same arc.

But we don't know for certain yet that this is always true for every single case.

In order to know for certain, we need to think about how we can prove this, and one way we can prove it is by using some algebra.

So Jacob is now going to prove that the angle at centre is always twice the angle at circumference for the angles subtended on the same arc by using a little bit of algebra and he's going to talk us through it.

He says, "Angle ABC, which is the one on the circumference, and the angle AOC, which is the one at the centre, are both subtended by the same minor arc AC.

This means that angle AOC should be double angle ABC." If we can prove that it's true.

He says, "If I draw in the radius OB, like this, I split the angle ABC, the one at circumference, into two parts.

I'll label these parts x degrees and y degrees.

Now you can see that by drawing the radius OB, I've also created two isosceles triangles." He says, "I can mark equal sides on my diagram like this." Each of those three sides is a radius of the circle, so they're all equal.

That means we now have two isosceles triangles and we can mark other angles on the diagram.

So for example, in this triangle we have x degrees, we also have this other angle is x degrees as well.

Angle OCB is equal to angle OBC.

They're both x degrees because base angles of isosceles triangle are equal.

Now we have an expression for two of the angles.

We can write an expression for the third angle by using the fact that those three angles should sum to 180 degrees.

So that means an expression for the angle at the centre is 180 degrees, subtract two x degrees.

And then we can do the same for the triangle on the right with y degrees.

And now when we look at the angles at the centre of the circle, we have three angles around that point in the centre.

We have expressions for two of them, and we want to write an expression for the third angle.

Those three angles sum to 360 degrees.

So if you want to write an expression for the missing angle, we could do 360 degrees, subtract each of the other two angles that we have expressions for.

It would look something a little bit like this.

Let's remove some of this information now and focus just on the key ones, these.

The angle at circumference, ABC, is equal to the sum of x degrees and y degrees.

The angle at the centre, AOC, is equal to the sum of two x degrees and two y degrees.

We could factorised that to make it two times the sum of x degrees and y degrees.

And that is two times what we have for the angle at the circumference.

Therefore the angle at the centre, AOC, is equal to two times the angle at the circumference, ABC.

So if it works when we use variables instead of actual numbers, it means that it always works.

So Jacob says, "This serum works for any angle." Okay, so it's off now to Task B, this task contains three questions and here is question one.

You need to calculate the missing angles.

Pause the video while you do it and press play when you're ready for more questions.

And here are questions two and three.

Pause the video while you do that and press play when you're ready to look at some answers.

Here are the answers to question one.

Pause while you check these against your own and press play for more answers.

Here are your answers to question two.

Pause while you check and press play to continue.

And here is a work solution for question three.

Pause while you read through this and it against your own and then press play when you're ready to continue.

Fantastic work today.

Now let's summarise what we've learned.

We've learned about theorems. A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments.

Theorems can be thought of as puzzles to solve and you are showing how to solve it.

We've also focused particularly on one circle theorem, and that is the angle at the centre of the circle is always twice the angle of circumference at any point of the circumference, so long as they're both subtended by the same arc.

And then we've learned how to use that theorem and you can use that theorem by working out missing angles, but you may need to sometimes draw a diagram or add lines onto the diagram yourself in order to see how you work out missing angles.

Well done today.

Have a great day.