Lesson video

Hi, I'm Mr. Chan, and in this lesson, we're going to look at some mixed circle theorem problems. Let's begin by looking at some of the circle theorems that we should already know.

So, in the first example, we've got an angle at the centre, and an angle at the circumference, but how we look out for these is to notice that both of these angles have been subtended by the same arc.

What that means is they've both been drawn from the same arc.

So what we can say about this circle theorem, and what links these two angles is, the angle at the centre is twice the angle at the circumference.

In this next example, we've got a diameter.

We know it's a diameter because it's a straight line that goes through the centre, at an angle coming off the diameter.

So, what the diameter's done is it's created a semicircle in the circle, so it's two equal semicircles.

Now the diameter there, the angle at the circumference has been subtended from the diameter.

So what we can say about the angle in the semicircle is that it is 90 degrees.

Let's look at some more circle theorems we should know.

So in this circle theorem, what we can see is that these angles here have all been subtended by the same chord.

Now that chord is generally not drawn for you, but the important thing with this chord is that the chord has split the circle up into two segments.

So, all the angles on the right hand side of that chord are in the same segment.

Now, when we have angles subtended by the same chord, so all of those angles have been drawn from that same chord, what we can say about angles in the same segment, they are equal to each other.

Another circle theorem that we should know.

Here, we have a cyclic quadrilateral.

Remember, a cyclic quadrilateral is a four-sided shape where all the vertices are on the circumference of the circle.

So that is a cyclic quadrilateral.

What we can say about the angles inside the cyclic quadrilateral are that the opposite angles are always going to add up to 180 degrees.

Here's another circle theorem that we should know.

This circle theorem, we can see there's a tangent, and a radius, and what we see is notation there that's boxed off to indicate that it's a right angle.

So this circle theorem tells us that a tangent and a radius are perpendicular at the point of contact.

So that just means they meet at a 90 degree angle.

The next circle theorem to review is that the angle between a tangent and a chord is equal to the angle in the alternate segment.

This circle theorem is known as the alternate segment theorem.

So that tells you that the angle that a chord and a tangent makes will be equal to the angle subtended by the same chord in the other segment.

Just one last circle theorem to review before we begin some questions.

In this circle theorem, we can see that there's a perpendicular drawn from the centre of the circle to a chord.

Now what happens with this circle theorem, it tells us that the perpendicular from the centre to a chord will bisect the chord.

So the chord has been cut into two equal halves.

Here are some questions for you to try.

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

Here are the answers.

So in these mixed circle theorem problems, you're going to probably have to use more than one circle theorem to answer the questions.

Part B is a really good example of that, where you're going to have to use the circle theorem that the angle at the centre is twice the angle at the circumference, and also, knowing that that chord is bisected with a perpendicular from the centre.

That creates a 90 degree angle, so you can figure out what angle X and Y are, in that question.

Here are some questions for you to try.

Pause the video to complete the task.

Resume the video once you're finished.

Here are the answers.

How did you get on? The secret to part A is realising that there are some radii drawn, so that creates isosceles triangles.

And in part B, knowing that you're going to use that the angle at the centre is twice the angle at the circumference, and also the alternate segment theorem.

Here's another question for you to try.

Pause the video to complete the task.

Resume the video once you're finished.

Here are the answers.

Hopefully you got this correct.

Now, if you didn't quite get there, the clue to this question is figuring out the angles that you can.

So, try starting with angle ADM, and then once you've got that angle, think about the information that you've been given, that CD is a diameter.

Think about the angle subtended from a diameter, review the examples at the beginning of the lesson, if you get stuck, Here's another question for you to have a go at.

Pause the video to complete the task.

Resume the video once you're finished.

Here's the answer for question four.

It's pretty clear that it's a cyclic quadrilateral.

And what do you know about cyclic quadrilateral angles? Yes, the opposite angles in a cyclic quadrilateral sum to 180 degrees.

So with this question, I can see that we have an angle 3X subtract four, and also an angle, opposite, 2X plus 24.

So I can add those two together, make them equal to 180, and we have an equation we can solve, that will give you the value of X, and once you've got that, you can substitute that into the angle 2X, and you can find angle Y from that point on.

That's all for this lesson.

Thanks for watching.