Lesson video

Hi, I'm Mr. Chan.

And in this lesson, we are learning about the circle theorem, the angle in a semicircle is 90 degrees.

Let's begin with this example.

We're asked what is the size of angle x? And how do you know? from another circle theorem, we know that the angle at the centre, is twice the angle of at the circumference.

So the angle the centre is 140 degrees.

So that must mean, the angle x is 70 degrees.

This example, asks again, what is the size of angle x? And how do you know? Well in this diagram, we have a diameter, I know also the diameter, because it's a straight line that goes through the centre of the circle.

So that means that the angle of the centre in this case would be 180 degrees.

Because it's a straight line.

So from the circle theorem, the angle at the centre is twice the angle the circumference, that must mean that x is 90 degrees.

Let's look at this circle theorem in more detail.

So when we have a diameter, that splits the circle up into a semicircle, what we will find, is that the angle in the semicircle is 90 degrees.

Because it doesn't matter how we rotate the diameter, the angle that's that's created at this circumference will always be 90 degrees, because the angle of the centre is always double the angle of the circumference.

So if you've got 180 degrees at the centre, the angle the circumference must always be 90 degrees.

So this is where we get the circle theorem.

The angle in a semicircle is 90 degrees.

In this example, we're asked to find the size of angle x.

And again asked, how do you know? So I can see a diameter, which means we can use the circle theorem, angle in the semicircle is 90 degrees.

So the angle the circumference must be 90 degrees.

We also know, that the angles in a triangle sum to 180 degrees.

so if we calculate angle x by subtracting 90 and 67 from 180, we get a value x equals 23 degrees.

Here's a question for you to try.

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

Here are the answers, in these questions you're really practising how to identify the angle and semicircle is 90 degrees.

So what you're looking for is the diameter and the angle subtended by the diameter of the circumference is 90 degrees.

So hopefully you got all those correct.

Here's another question you can try.

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

Here are the answers for question two.

These questions rely upon you knowing that the angle in a semicircle is 90 degrees and then using that.

Also, be really careful in spotting isosceles triangles, where you know that the base angles are equal in an isosceles triangle.

So in part A, we have an isosceles triangle, as indicated by the dashes on those two lines in the triangle.

And also in part B, there are isosceles triangles, because the triangles are made up of radii.

Here's two problems you can have a go up, pause the video is complete the task, resume the video once you're finished.

Here are the answers.

So these questions again, they're using the circle theorem.

Angle in a semi circle is 90 degrees.

If we look at question four, we can see the angle in the semicircle about circumference is 90 degrees.

But then we've got to figure out angle x and angle y.

But we're told the ratio of angle x, to the ratio of angle y is two to seven.

So what we've got to do is share the 90 degrees that's leftover because angles in a triangle most add up to 90 degrees, and share that between angle x and angle y in that ratio, I will find the angle x equals 20 degrees and angle y equals 70 degrees.

Here's a proof question for you to try the examples that were covered at the beginning of the lesson.

We'll help you with this.

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

Here's the answer.

This question asks us to prove this circle theorem.

Angle in a semicircle is 90 degrees.

So we begin this proof by drawing a line from the centre to the circumference and what that does is create two isosceles triangles.

What we know about isosceles triangles is that base angles are equal.

So from there, we can then prove the circle theorem, the angle in the semicircle is 90 degrees.

That's all for this lesson.

Thanks for watching.