Lesson video
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Hi, I'm Miss Davies.
In this lesson, we're going to be drawing and recognising circle graphs.
Alex and Afia each think of a number.
They each share their number with their teacher.
Their teacher says, "The sum of the squares "of the numbers is equal to 25." What numbers could Alex and Afia have been thinking of? The possible solutions are three and four, four and negative three, zero and five, negative three and negative four, and negative five and zero.
There are more possible solutions than just these five.
Looking at these five sets of numbers, do they fit a pattern? Can we visualise these solutions? Could we plot them as coordinates? Let's plot Alex's number as the x-coordinate, and Afia's number as the y-coordinate.
So the possible coordinate pairs would be three/four, four/negative three, zero/five, negative three/negative four, negative four/three, and negative five/zero.
What if we plotted more of the possible solutions as coordinates onto this grid? We've now got another six possible solutions.
This has formed a circle when we've joined up all of the possible solutions.
This circle has a radius of five, and a centre zero, zero.
What do you think the equation of the circle is? Their teacher said that the sum of the squares of the numbers is equal to 25.
We said that Alex's number was the x-coordinate, and Afia's number was the y-coordinate.
If we switch those around it would still give us 25, as addition is commutative.
This 25 relates to the radius of the circle, squared.
We can generalise this equation as the equation of the circle is x squared, add y squared is equal to the radius squared.
Looking at this next example, we've been asked to write down the equation of the circle A.
We can see that the centre of this circle is zero/zero.
The radius of this circle is three.
So the equation of circle A is x squared, add y squared is equal to three squared.
Or x squared, add y squared, is equal to nine.
Let's have a look at this next example.
We can see from our circle that the circumference sits halfway between three and six, on both the x and the y axes, as well as halfway between negative three and negative six.
Halfway between three and six, or the midpoints of three and six, is 4.
5.
This circle has a centre of zero/zero.
This means that the equation of circle A is x squared, add y squared, is equal to 4.
5 squared, or x squared, add y squared, is equal to 20.
25.
Here are some questions for you to try.
Pause the video to complete your task, and resume once you're finished.
Here are the answers.
You might have written these as circle one, x squared, add y squared, is equal to 10 squared.
Circle two, x squared, add y squared, is equal to 7.
5 squared.
And circle three as x squared, add y squared, is equal to 13.
75 squared.
Here are some questions for you to try.
Pause the video to complete your task, and resume once you're finished.
Here are the answers.
On part b, the radius is seven, as the diameter is two times the radius.
And on part c, the radius is 10, as the area of the circle is pi times the radius squared.
Here is a question for you to try.
Pause the video to complete your task, and resume once you're finished.
Here is the answer.
The radius is six, as to find the circumference of a circle, you multiply pi by the diameter.
And the diameter is twice the size of the radius.
Here is a question for you to try.
Pause the video to complete your task, and resume once you're finished.
Here is the answer.
To find the radius, we need to do the square root of 42.
25, which gives us 6.
5.
We can then state that the diameter is 13, meaning that the circumference is 13 pi.
Our next question is asking us to sketch the graph of x squared, add y squared, is equal to 16.
From this equation, we know that the centre of the circle is zero/zero, and the radius is four, as the square root of 16 is four.
This means that our circle will look like this.
Here are some questions for you to try.
Pause the video to complete your task, and resume once you're finished.
Here are the answers.
The radius of the first circle is seven, and the radius of the second circle is 2.
5.
This example tells us that the equation of a circle is given as x squared, add y squared is equal to a squared, where a is the radius of the circle.
C2 is a concentric circle to C1, that has half the area of C1.
We have been asked to write the equation of C2 in terms of x, y, and a.
Concentric circles share the same centre.
This means that C2 is in the equation x squared, add y squared equals the radius squared.
Circle two has half the area of circle one.
The area of circle one is pi a squared, as a is the radius.
This means that the area of circle two is pi a squared over two.
This tells us that radius squared in circle two is a squared over two.
This means that the equation of C2 is x squared, add y squared is equal to a squared, add two.
Here is a question for you to try.
Pause the video to complete your task, and resume once you're finished.
Here is the answer.
The radius squared of the second circle is two a squared over three, as the area is two thirds the area of the first circle.
That's all for this lesson.
Thanks for watching!.
