Lesson video

Hi, I'm Mr. Chan.

And in this lesson, we're going to learn about the circle theorem, a tangent and a radius are perpendicular at the point of contact.

Let's begin looking at this circle theorem.

This circle theorem looks at the angle between a tangent and a radius at the point of contact.

So let's draw a tangent.

The tangent is just simply a straight line that touches the circle at one point, and we draw a radius to meet the tangent.

And what we find is that the angle, that these two lines create the tangent and the radius is a 90 degree angle, and that's the right angle.

And what happens if we draw the tangent at different points? What you see is it doesn't matter where the tangent's drawn.

If a radius is drawn to meet the tangent, it will always be 90 degrees.

So what we can say about this circle theorem is, a tangent to the circle meets the radius at 90 degrees.

Let's look at an example where we can use this circle theorem.

Here we have to work out the size of the angle marked x.

And what we can see is we've got a tangent and radius.

And this circle theorem tells us that the tangent meets the radius at 90 degrees.

So the angle where the tangent meets the radius is a right angle at 90 degrees.

What we also know is the x is inside the triangle.

So the angles in the triangle, we know sum to 180 degrees.

So to work out the size of angle x, we simply subtract the 90 degrees and the 39 degrees from 180, to get an answer of x is 51 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.

You can see that you need to use this circle theorem in order to work out angle x.

So that you know that the angle between the tangent and the radius is 90 degrees.

And then to figure out angle y, Angle y is in a triangle, so you can work out angle y by subtracting 90 degrees and the 43 from 180, that will help you get the angle y is 47 degrees.

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.

Always be on the lookout for special notation on your diagrams, especially where you see dashes on lines.

Those dashes indicate that the lines are equal in length.

And in the case of triangles, where you've got two triangle lengths are equal in length, that makes a triangle isosceles and base angles in an isosceles triangle are equal.

That will help you figure out some of these angle questions.

So in the case of question B, C and D, you can see that there are isosceles triangles there.

So base angles in an isosceles triangle equal, that will help you with those questions.

Let's look at another circle theorem that relates to tangents and the circle.

What we've got here is a point exterior to the circle.

And we're drawn to tangents to the circle, as you can see.

Now let's measure one of those tangents.

We can see that this bottom tangent there is seven centimetres long.

And if we measure the other tangent from the point to the point of contact, that would also be seven centimetres long.

So what we can say about these two tangents is that they are equal in length, and we can show that by drawing these dashes on the line.

And what we find is that if we always draw the tangent from an exterior point and measure them, they will always be equal in length.

So wherever the point is, they will always be equal in length.

So this circle theorem tells us that two tangents from a point are equal in length.

Here's a question for you to try, pause the video to complete the task, resume the video once you're finished.

Here's the answer to question three.

You can see that both of those tangents are drawn from the same point, point A.

So that means the tangents are equally length.

So if AB is 12 centimetres, that must mean AC is also a 12 centimetres.

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.

So in part a you can figure out the length of AC by realising that the two tangents drawn from the same point.

So the two tangents must be equal in length.

And in part b, what you do see is the radius and the tangent meeting at one particular point of contact.

So that must mean that the angle that is 90 degrees and what that creates is a 90 degree angle, a right angle triangle.

So that means you can use Pythagoras theorem in order to figure out what length AO is.

So you've got a side length of seven centimetres, another side length of 24 centimetres.

AO would be the hypotenuse.

So using Pythagoras' theorem, you can figure out length AO equals 25 centimetres.

That's all for this lesson, thanks for watching.