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Hi, I'm Mr. Chang.

And in this lesson, we're learning about the circle theorem, where the perpendicular from the centre to a chord bisects the chord.

Let's begin by looking at this example.

I'm going to draw a line AB.

Now AB is chord because it joins two parts of the circumference together.

And if I draw a perpendicular from the centre to the chord, so perpendicular means those two lines must meet at right angles.

What I find is that point C has actually bisected line AB.

So what we can say is that AC equals BC.

And what we find is that doesn't matter where the chord is drawn, that that will be the case that a perpendicular from the centre to the chord will actually bisect the chord into two equal halves.

Let's look at an example where we can put this circle theory into practise.

So we know the perpendicular from the centre to a chord bisects the chord.

And in this example, we're told the AC equals 12 centimetres.

And we're asked what is the length of BC? Well, we can see the AB is chord, and OC is a perpendicular from the centre to the chord.

So that means C must be the midpoint.

So if AC and BC are equal in length, that must mean the BC equals 12 centimetres.

Here's another example.

In this question we're told AB equals 36 millimetres, what is the length of BC? So, again we can see there's a chord drawn which is AB, we can see a perpendicular from the centre O to C.

So because that's a perpendicular that means the chord has been bisected.

So that means in terms of working out what BC is that's half the length of the chord we know the full chord is 36 millimetres that must mean that the length of BC is equal to the length of AC.

So we're splitting the AB length up into two equal parts.

That means BC must be half of 36, which is 18 millimetres.

Let's look at another example.

So in this example, we're told the radius of the circle is 10 centimetres.

The chord AB is 16 centimetres, and we've got to work out the length CO.

So the radius is 10 centimetres.

Let's draw that in and I'm going to draw a line from O to B, which represents the radius of 10 centimetres.

Now if we look at the triangle OBC, I can see that's a right angled triangle because on one side of the chord, and the perpendicular is right angle, so the other side must be a right angle as well.

And we're also told AB equals 16 centimetres.

And because the line OC is a perpendicular, we know that, that must bisect the chord into two equal halves.

So that must mean that if AB is 16 centimetres, BC, must be half of that, which is eight centimetres.

What we have now is a right angle triangle with one side being eight centimetres, the hypotenuse of 10 centimetres.

We can use Pythagoras to work out the length of CO.

So let's do that.

So let's put that into Pythagoras' theorem.

Eight squared plus CO squared equals 10 squared.

Figure out what eight squared and 10 squared are.

And we will find that CO squared equals 36.

To find CO we square root that so we find CO to be six centimetres.

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.

So in this question, you're told the OM is perpendicular to the chord AB, the diagram tells you that AM is nine centimetres.

So the circle theorem does tell you that if OM is perpendicular that means it bisects the chord AB into two equal halves, If you given one half is nine centimetres, then the other half of the chord must also be nine centimetres.

So that's the answer.

Here's another 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 two.

In this question, the circle theorem helps us work out that BM is 21 centimetres.

And from there we have a right angled triangle where we can workout the length of OM using Pythagoras' theorem.

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.

This question tests circle theorem from a different perspective.

It tells you that the chord AB has to equal halves AM and MB because they're equal to each other.

So that means that the length of M must be a perpendicular, which means that it's a right angle triangle in terms of triangle O, B, and M.

If that's the right angled triangle, you've got another angle 48 degrees, you can work out the missing angle OBM which means then you can work out angle x because angles on a straight line sum to 180 degrees.

Here's another question for you to try.

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

Here's the answer.

In this question, we're told that OM is perpendicular to the chord AB.

So that means AB has been bisected and AB being nine centimetres, that means both parts of that chord must be 4.

5 centimetres each.

Now, in order to figure out length AC we need to know the length of CM.

Now we're not told what CM is, but we're told that the circle has a radius of nine centimetres and that the ratio CO to OM is three to two.

So using that ratio, we can figure out that CO being nine centimetres because that's the radius.

If that's three parts, that means one part must be three centimetres, the two parts must be six centimetres.

So that means that CM in total must be 15 centimetres.

If we know 15 centimetres is the height of that triangle, and AM is 4.

5 centimetres.

We can now use Pythagoras' theorem to figure out what the length AC is.

That's all for this lesson.

Thanks for watching.