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Hello, Mr. Robson here.
Welcome to Maths.
We're learning about ratios, inline segments on coordinate axis today.
We've seen ratios elsewhere in maths.
I wonder what they're going to look like here.
Should we take a look? I learning outcome is it'll be able to use ratios to divide a line segment and find the coordinates of a point on the line.
A key word we're gonna come across today is midpoint.
The midpoint of a line segment is halfway between the two endpoint.
Two parts to today's lesson, and we're gonna start with midpoint.
Let's go back to something you're very familiar with because it'll help us to understand today's concept.
Jacob and Jun are sharing 60 pounds in the ratio of one to one.
Let's look at that ratio one part to one part.
That's two parts, but we don't want two parts.
We want 60 parts because we're sharing 60 pounds.
We need to multiply everything by 30.
One times 30 and one times 30.
Jacob is getting 30 pounds and June is getting 30 pounds.
Jacob says, "I get 30 pounds.
That's half the money." Jun says, "Yes.
That's because in the ratio of one to one, we are each getting one outta two equal parts, which is a half." If you get this, you are going to get today's learning.
Line segment AB is 60 millimetres in length.
P is a point such that AP to PB is in the ratio of one to one.
That's our length AB and it's 60 millimetres.
AP to PB is in the ratio of one to one.
That's two parts.
We don't want two parts.
We want 60 parts.
One part for every millimetre we need to scale up by 30.
One times 30 and one times 30 gives us the ratio of 30 to 30 for AP to PB.
We can draw that 30 millimetres along from point A.
We find point P and from P to B is 30 millimetres also.
P is halfway along the line AB, we call this the midpoint.
The midpoint of a line segment is halfway between the two endpoints.
We can use this skill to find the midpoint of a line segment on a Cartesian coordinate grid.
In this example we might be told P is a point that AP to PB is in the ratio of one to one and then we might be asked to find P.
First, we can identify that length.
It's six.
We can see that from the coordinates.
We can read it from the grid.
AP to PB is in the ratio of one to one.
That's two parts.
We don't want two parts, we want six parts, so we scale up the ratio like so.
When we draw this on three to three in terms of the two lengths, we find point P the midpoint of the line, AP equals PB.
We have found the midpoint.
Quick check.
You've got this so far, which coordinate pair is the midpoint of line segment AB three choices there, pause, which one is it? Welcome back.
I hope you said it's B, 5.
5, seven.
The whole length is five.
Because the midpoint is halfway, we can do five divided by two.
It's two length of 2.
5.
AP is 2.
5, PB is 2.
5.
If AP is equal to PB in length, then P is a midpoint.
The line segment, which we're finding in the midpoint will not always be vertical or horizontal.
P is a point such that AP to PB is in the ratio of one to one and we're asked again to find P.
Because this example's different.
We're going to do it differently or is it the same thing? We're gonna consider the x journey to get from A to B.
The x journey from A to B is to move six in the positive x direction.
Then we're gonna consider the y journey from A to B.
We need to move four in the positive y direction.
What we then need to do is divide these journeys in the ratio of one to one.
In terms of that x journey, we'll turn a journey of six into two parts, three and three.
We'll do the same for the y direction.
We'll take a journey of four.
I'll turn it into two and two.
Once we've done that, we get this journey three in the positive x direction, two in the positive y direction.
That must be where P is located.
We know that P is the midpoint because these two triangles are identical.
Quick check.
You've got that.
Which coordinate pair is the midpoint of line segment AB in this example, there's three to choose from.
Pause and take your pick.
Welcome back.
I hope you said it's C, the coordinate pair four, four.
Why would it be that? Well, if we consider how we get from A to B, we need a movement of eight in the y direction and a movement of four in the x direction.
If it's the midpoint, what we want is half that movement, half of eight is four, half of four is two.
It's that movement to get from A to the midpoint P.
we know that P is the midpoint because the two triangles are identical.
Another quick check.
Now, how do we know that five two is not the midpoint of line segment AB in this case? What's different about this one? Pause, have a think.
Welcome back.
You might have said the triangles are not identical.
Therefore the ratio A to PB is not one to one.
Well done.
We can find the midpoint numerically.
We've had a lot of visual representation so far we don't necessarily need them.
We can find the midpoint numerically.
Find the midpoint of line segment AB whereby the coordinates of A and B are negative nine, one and one, negative five respectively.
In the x direction, to get from A to B, we're moving 10 in the positive direction.
We found that by doing the second x coordinate one minus the first x coordinate negative nine.
We need half that journey.
When we start at the first x coordinate of negative nine and move positive five, we would get to an x coordinate of negative four.
We need to repeat this for the Y direction to get from the first y coordinate to the second y coordinate, we need to move by negative six.
That's the second y coordinate negative five.
Subtract the first y coordinate one, but we don't wanna do the whole journey.
We wanna do half that journey, so we'd need to move from that first y coordinate of one negative three and we'd get to negative two.
The y coordinate will be negative two.
The midpoint is negative four, negative two.
You can do that problem numerically, but I find it really powerful if we visualise the problem.
A simple sketch helps us to do that.
The exact same problem supported by a sketch negative nine, one is in the second quadrant.
One negative, five is in the fourth quadrant and what we want is to do half this journey.
We know to get from negative nine to one in the x direction, we move by 10 and to get from one down to negative five in the y direction, we move by six.
But we only want half that journey, so we just move three and five.
Negative nine add five in the x direction takes us to negative four and in the y direction.
When we start at one and we go down by three, we get to negative two.
P, the midpoint must be a negative four, negative two.
I find that problem much easier to understand and I've got a visual representation of what's going on.
Quick check.
You've got that.
Find the midpoint of line segment AB whereby the coordinates of A and B are negative two, negative three and 10, five respectively.
You are welcome to do this numerically, but my hint is draw a sketch, pause, have care this problem now.
Welcome back.
Let's see how we got on.
I start with a sketch.
Coordinate A is in third quadrant, coordinate B is in the first quadrant.
I need half this journey.
In the x direction, we're travelling 12.
In the y direction, we're travelling eight.
I need half of each of those journeys.
We're gonna move six in the x direction, four in the y direction.
That takes us to four, one coordinate.
P, is at four, one.
We know this is the midpoint because an identical triangle takes us to point B.
Jacob and Jun are discussing this problem.
P is a point such that AP to PB is in the ratio of one to one and we're asked to find B.
Jacob looks at the line and says, "That's easy.
The midpoint of that line is three, three." Jun says, "Not so sure, Jacob.
I don't think it's that straightforward." Who do you agree with you with Jacob? The answer's three, three.
Are you with Jun? There's a little bit more to this problem.
Pause, have a conversation with the person next to you or a good think to yourself.
Welcome back.
I wonder what you think.
Jacobs realised something.
Of course, Jun P is the midpoint.
B is the other end of the line segment.
Jun says, "Yes.
We can use identical triangles to find it." If that's the journey from A to P, then the journey from P to B is the exact same because they're in the ratio of one to one.
So we find coordinate B at nine, six.
Jacob says, "So B is nine, six." And Jun says, "Yes." The ratio helped us to see that B wasn't end point.
Practise time now.
Question one.
I'd like you to find the respective midpoint, the midpoint of line segment AB, the midpoint of line segment BC, and the midpoint of line segment AC, pause and do that now.
Question two.
P is a point such that AP to PB is in the ratio of one to one, the coordinates of A are 32, negative 18 and the coordinates of B are 50, 100 find the coordinates of P.
My hint is that you draw a sketch to support yourself on this problem.
Question three looks incredibly similar, but read it carefully because the very important difference.
Pis a point such that AP to PB is in the ratio of one to one.
The coordinates of A are two, eight.
The coordinates of P are five, 20.
Find the coordinates of B.
Again, my hint is that you draw a sketch to support your learning.
Pause and give this problem a go now.
Feedback time.
Question one.
We're asked to find the respective midpoint for A, we're asked to find the midpoint of line segment AB.
That would be three, four.
For B, the midpoint of line segment BC was there 0.
5, zero.
You might have written a half zero.
That would be a perfectly acceptable answer.
For C the midpoint and line segment AC is there.
That's negative 1.
5, one.
For question two, I suggested a sketch would help to support the learning on this one.
It certainly helped me when I did this problem.
Quadrant A 32, negative 18 is in the fourth quadrant.
Quadrant B 50,100 is in the first quadrant and we can see that the movement in the x direction is positive 18.
The movement in the y direction is positive 118 If we're going to find the midpoint, because if AP is equal to PB, P is the midpoint, then we just need half that journey.
I wanna move nine in the positive x direction and 59 in the positive y direction.
That would take me to that point there.
P must be at 41, 41.
Question three, I hope you read this one very carefully.
AP to PB in the ratio of one to one means our journey from A to P is the same as our journey from P to B, so B must be an eight, 32.
Onto the second half of the lesson now, dividing in a given ratio.
The ratio which we're dividing a line segment will not always be one to one.
Line segment AB is 60 millimetres in length.
P is a point such as AP to PB is in the ratio of two to three.
Let's start by writing the length 60 millimetres.
In the ratio of two to three.
We're talking about five parts.
We don't want five parts, we want 60 parts, so we multiply through by 12.
Two lots of 12, three lots of 12.
If AP is 24 millimetres, then P lies here and PB is 36 millimetres.
P is no longer the midpoint.
We've divided the line segment AB in the ratio of two to three.
We will encounter similar ratio problems on Cartesian coordinate grids.
P as a point such that AP to PB is in the ratio of one to three.
Find P.
We know that P is not going to be a midpoint.
Because the line segment is being divided in the ratio of one to three, P will be one quarter the length from point A and it'll be three quarters the length from point B with the ratio one to three, we've got four parts so it makes sense to define line segment AB into four parts, which case P must be there.
Can you see how AP to PB is in the ratio of one to three? We will see more complex examples than that one.
P is a point such that AP to PB is in the ratio of one to three or asked again to find P.
P is still a quarter the length from point A.
It's just not so easy to see in this example, but we know the length of the journey in the x direction and the length of the journey in the y direction.
We need to travel positive 12 in the x direction and positive four in the wide direction.
We can divide these lengths in the given ratio of one to three.
When we do that, we turn our journey of 12 into a journey of three to nine and our journey of four in the y direction into a journey of one and three.
Let's do that visually on the grid, three to nine in the x direction, one to three in the y direction when we arrange it like so we've solved the problem.
What we've got here is two pairs of size in the same scale factor one to three, that's a scale factor of three, three to nine in a scale factor of three, we've got the same angle, right angles.
These are similar triangles.
We found P at coordinate negative seven three.
Quick check.
You've got that.
P as a point such that AP to PB is in the ratio of two to three.
I'd like you to find point P.
Pause, give this problem a go now.
Welcome back.
The journey from A to B means moving 15 in the y direction, 10 in the x direction.
We can divide those journeys in the given ratio.
Our journey of 10 in the x direction will become a journey of four in a journey of six are 15 in the y direction it'll become a journey of six and a journey of nine.
See what that looks like visually and then we'll arrange that like so.
That must be point P at coordinate negative three, zero and again you can see two similar triangles six to nine is in the ratio of two to three, four to six is in the ratio of two to three.
That's how we know we're right.
Alternatively, in the ratio of two to three, we have five equal parts, so what would one equal part look like? Well, it would involve dividing each journey into five parts.
One part of the whole journey will look like that and if we complete five of that journey, we get all the way from A to B, so point P must be at that moment because I can see five parts with AP having two parts and PB having three parts.
Jacob and Jun are discussing this problem.
Point A is at one, one point B is at seven, four.
Part A of the problem asks find point point P when AP to a B is in the ratio of one to two.
For part B.
Find point P when AP to PB is in the ratio of one to two.
Jacob says.
"AB look like the exact same problem, but I suspect they are different.
Can you help me June?" That's awesome, Jacob.
We're all gonna get a little bit stuck in mathematics and when we do, it's really good that we ask our peers for a little help.
What if Jacob was asking you what advice would you give about this example? Pause.
See if you can come up with some good advice now.
Welcome back.
Jun's got some really good advice.
Jun says, "There are lots of ways we can do this, but I think of visual representation.
We'll help get us started." For part A, A is at coordinate one, one, B is at coordinate seven, four.
That's the length AB and that's the length AP, y.
Jun spotted it.
For problem A, we can see that AB is twice the length of AP.
We're told that by the ratio, the ratio of one to two tells us that AB is twice the length of AP.
Therefore P must be the midpoint.
There's P and on this occasion it's the midpoint.
For part B, the problems ever are slightly different.
AP to PB with the line segment AB divided up like so and PB being there.
Why is that? Well, Jacob sees it now.
Thanks to Jun's good advice of using a visual representation.
Jacob says, "For B, we can see that the journey from P to B is twice that of A to P." You can see in the ratio a P to PB in the ratio of one to two.
Therefore, P divides the line segment in that ratio.
Thanks for your help, Jun.
Well done both of you I say.
Quick check.
You've got this now.
Point A is at one, negative five point B is at four, four.
Select the point representing point P when AP to PB is in the ratio of one to two.
You've got three choices there.
Pause and consider which one is the right one.
Welcome back.
I hope you went for C.
The length PB is twice that of AP.
Another check now.
I'd like you to select the point representing point P when AP to AB is in the ratio of one to two.
Again, three choices, pause and I hope you make the right one.
Welcome back.
I wonder which one you went for.
Hopefully option B, the whole length AB is twice that of AP.
Therefore P must be the midpoint.
Practise time now.
Question one.
Point A is at negative two, one, point B is at seven, four.
Find point P when AP to PB is in the ratio of one to two.
Pause and try this problem now.
The question two point A is at negative eight, negative six, point B is at six, one.
Find point P when P to PB is in the ratio of four to three.
Pause and try this one now.
For question three.
Point A, is that negative seven, eight, point B is that two, negative four.
For part A, I'd like you to find point P when AP to AB is in the ratio of one to two.
And for part B, find point P when AP to PB is in the ratio of two to one.
Pause and try this problem now.
Question four.
Point A, is it negative seven, eight.
Point P is it two, negative four.
Find point B when AP to AB is in the ratio of three to five, you can do this one numerically, but my hint is that you draw yourself a sketch to help your learning.
Pause give this problem a go now.
Feedback time now.
Question one should have found that point P, is at one, three.
How do we know? Because we've got two similar triangles with a scale factor of two or ratio of one to two, the base lengths three to six, that's a ratio of one to two.
Scale factor of two, the heights and a ratio of two to four.
That's the same ratio as one to two or it's a scale factor of two.
We know we're right because the ratio of the lengths between those two triangles are in the ratio of one to two.
Question two we're asked for point P when AP to PB is in the ratio of four to three.
To get from A to B, we need to travel 14 in the x direction and seven in the y direction.
We have seven equal parts.
What would one part look like it would look like? So if we take that journey and do it seven times we'll get from A to B.
That visual representation is really powerful.
It enables us to say that point P is there because I've got four parts between A and P and three parts between P and B.
Point P is at zero, negative two.
Question three, part A, find point P when AP to AB is in the ratio of one to two.
P would be at the coordinate negative 2.
5, two because the whole length AB is twice that of AP.
Therefore P is a midpoint.
For part B, P had a coordinate negative one, zero.
AP is twice the length of PB on that occasion.
We had to pay particular attention to the wording and detail in the question in order to get those two right.
For question four, recommended a visual representation which would look something like that.
We were asked to find point B when AP to AB is in the ratio of three to five.
What we want to do is turn that AP into three parts because we know AP is three parts to AB's five parts, so we make AB, five of these parts.
B must be there at the point eight, negative 12.
That's the end of the lesson now and we've learned we can divide a line segment by a given ratio to find the coordinates of a point on the line.
If the ratio of AP to PB is one to one, then P is the midpoint of line segment AB.
I hope you've enjoyed this lesson and I shall look forward to seeing you again soon for more mathematics.
Goodbye for now.
