Lesson video

Hi, I'm Mr Chan.

And in this lesson, we're going to look at the circle theorem that relates the angle of the centre with the angle at the circumference.

Let's begin by looking at an example.

Here, we have two triangles drawn inside a circle that share one common point being at the centre and also one common side being the radius.

In fact, both of these triangles are made up of radii.

So that makes both of these triangles isosceles.

And what we know about isosceles triangles is that base angles in an isosceles triangle are equal.

So that allows us to fill in both of those base angles for both of the triangles.

That now allows us to find the angles at the centre for both triangles, because we know angles in a triangle add up 180 degrees.

So that angle there must be 96.

And the other angle at the centre must be 144 degrees.

That then allows us to find the other angle at the centre because we know angles around the point, add up to 360 degrees.

The angle at the centre there must be 120 degrees.

Now what's interesting is the relationship between the angle at the centre made exterior to the triangle, 120.

And also the angle at the circumference, made by both of those triangles, which is the 42 and the 18 degrees at the top there.

So let's have a look at those in particular.

What do you notice? Let's have a look at another example.

Again, we've got two triangles that share a common Vertex at the centre and are made up of radii to make them isosceles.

So we can complete the missing angles at the base, 22 degrees and 28 degrees.

So the angles at the centre of the triangles must be 136 degrees and 124 degrees because the angles must add up to 180.

So the angle at the centre made by those triangles, exterior to the triangles must be a 100 degrees.

So the angle we're also interested is the angle at the circumference that the two triangles share, that's the 22 and the 28.

So if we add those together, we get 50 degrees.

So again, we've got an angle at the centre of 100, an angle at the circumference of 50 degrees.

What do you notice? Let's look at this circle theorem in a little more detail.

So the examples have shown we have an angle at the conference, and also an angle at the centre, and there's a special relationship between those two.

You can see one is twice the value of the other.

Now it's really important to know where these two angles have been drawn from.

And we call that subtended by.

And where these angles have been subtended by is from the arc that'S drawn on the diagram there.

So I've highlighted the arc in red there.

So the angle at the centre has been subtended by the arc.

So, what I mean is the angle that's been drawn from the ends of the arc to create an angle at the centre.

And the angle at the circumference has also been subtended by the same arc.

So that's what this circle theorem is really about.

It's saying that the angle at the centre is twice the angle at the circumference, subtended by the same arc.

And you can try and spot those when you're solving your own problems. When we're asked for reason, in order to write down a reason, this is what we would write down.

If we were ever asked, why is that angle that value? So we would say, the angle at the centre is twice the angle at the circumference.

And another way we could say this is the angle at the circumference is half the angle at the centre.

Here's some questions for you to try.

Pause the video to complete the task, resume the video once you're finished.

Here's the answers for the first question.

You'll notice that they all have something in common.

Yes, all the angles that you're looking for, are 60 degrees.

Hopefully question c didn't catch you out.

And you'll have noticed that the angle at the centre and the angle at the circumference have both been subtended from the same arc.

So I know the arc isn't drawn there, but hopefully you spotted that.

Here's some more questions for you to try.

Pause the video to complete the task, resume the video once you're finished.

Here are the answers.

So again, we're using the circle theorem that the angle at the centre is twice the angle at the circumference.

And the only question that may have caught you out again is part d.

Those two angles there are subtended from the same arc, there's an angle at the centre, there's an angle of this conference, so again, the angle at the centre is twice the angle at the circumference.

Here's some more questions for you to try.

Pause the video to complete the task, resume the video once you're finished.

Here are the answers.

I hope part d didn't catch you out.

In part d you're given the reflex angle, that's on the other side of the angle that's important for this circle theorem.

So have another look at that if you didn't get that right.

Here's a proof question that you can have a go at.

Think about the examples that we looked at the beginning of the lesson to help you.

Pause the video to have a go, resume the video once you're finished.

In this question, you're asked to prove the circle theorem that the angle at the centre is twice the angle at the circumference.

Now a lot of this proof all depends on many of the examples that we covered at the beginning of this lesson.

So have a look if you've missed any steps out.

That's all for this lesson.

Thanks for watching.