Lesson video

Hello and welcome to another video.

In this lesson, we'll be looking at plan problems. Now for this lesson, you're going to need a pen or a pencil, a ruler, and something to write on.

So go get those things before you get on with today's lesson.

Okay, now that you have all those things, let's get on.

Remember, my name is Mr. Maseko.

First try this activity.

So draw a rectangle and label the sides A, B, C, and D clockwise as is shown on the diagram, then bisect the angle BAC and find the perpendicular bisector of the line AD.

Pause the video here and give this a go.

Okay, now that you've tried this, let's see what you should have come up with.

Well, we want a perpendicular bisector of the line AD, let's do that first.

Well, we don't have the line AD here, so I'm going to draw it in.

Yours is going to be drawn with a straight line.

So I'm just going to draw mine roughly, there we go.

There's a line AD.

Now a perpendicular bisector, remember we first need the midpoint of that line and another point that's equidistant.

And the way you can find this is you can just join the point A to the point D and then your forward line, you will see a foward line and that forward line is your perpendicular bisector, and there we go.

And then you want to bisect the angle BAC so this angle here.

So how did we bisect the angle? Well, what did we do? Well, we got a point on that angle and then what did we do? Once we got the point on the angle, we joined those two lines together, and then we drew a straight line that bisected that angle.

Now your picture, what it should look like in an accurate sense is something that looks like this.

So this is our angle bisector, and this line was the perpendicular bisector of the line AD.

So Anthony says any point in region one, so in that region there is closer to A than to D, it is also closer to AC than AB.

Now what other statements can you make? Well, let's look at region two.

What statement can you make about region two? Well any point in region two is closer to AB, so closer to AB than what? AC.

How do we know? Because if you look, that angle bisector means that anything below that line is closer to the line segment AC.

So region two is closer to AB than AC and is also closer to what? It's closer to the point B than D, and it's also closer to the point A than D, that should say D.

Well, how do you know it's closer to A than D? Well, because this here is the perpendicular bisector of the line AD so anything above that line is closer to A than it is to D.

Coincidentally, it's also closer to B than it is to D.

Nextly, what about region three? Well in region three, you can see that we are closer to the line BD than we are to the line AC.

And we are also closer to point D than we are to point A.

Why? Because we are below the perpendicular bisector of the line AD.

And you can make those same kinds of statements for region four.

Now, here we are.

So let's explore this idea of finding regions closer to specific points a bit more.

Now we want to shade in the region that's closer to point A than point B.

How would we do this? Pause the video here and give this a go.

Okay, now that you've tried this, this is what you should have done.

Well, you should have first drawn a straight line that connected A and B and you want the region that's closer to A than B.

So then you'd have drawn your perpendicular bisector.

And remember how we did that, you could just take your sheet of paper and just join the two ends of your line together and that forward line that you get is your perpendicular bisector, it's cut your line in half.

And any point on this side of the perpendicular bisector is closer to A than it is to B.

Any point on the perpendicular bisector is equidistant from those two points, so we have to be on one side of the perpendicular bisector 'cause we have to be closer to A than B.

And then on this side of the perpendicular bisector, we are closer to B.

So what you have to know is that if you're trying to find regions that are closer to a specific point than it is to another, you have to draw a perpendicular bisector because you are trying to find a region that's closer to one point than it is to another.

But if you're trying to find a region that's closer to one line segment than it is to another, what do you have to do? This is what we have to do here.

We want to find a region that's closer to AB than it is to AC.

Pause the video here and give this a go.

Okay, now that you've tried this, let's see what you've come up with.

Well, the first thing you should have done is you should have constructed an angle bisector.

And when you bisected that angle, this is what should have happened, and that's your angle bisector.

So the region that's closer to AB than it is to AC is any point above that angle bisector, so that's shaded in region? So you see the difference? When we're trying to find a region that's closer to one line segment than it is to another line segment, when those two line segments meet together to make an angle, we have to construct an angle bisector.

So say it again, to find a region closer to one point, one point than it is to another, we construct a perpendicular bisector.

When we want to find a region that's closer to one line segment than it is to another, we construct a angle bisector, really well done.

So in this task, we want to shade in a region that's closer to D than it is to C and closer to the line segment CD than it is to the line segment BD.

Pause the video here and give this independent task a go.

Okay, now that you've tried this, let's see what you should have come up with? Well, to find a region that's closer to D than it is to C 'cause those are two points, we have to draw a, good, perpendicular bisector.

So we just draw a perpendicular bisector of the line CD, that's the first thing.

And then to find a region that's closer to CD than it is to BD, we have to construct what? An angle bisector.

Good and when we construct an angle bisector, this is what happens.

So a region that is both closer to D than it is to C and closer to CD than it is to BD and that would be this region here.

Why can't we be in that region? Well because in that region, we're closer to C than we are to D but we are closer to CD, but we can't be in that region because it's not closer to D than it is to C.

So the only region that satisfies that rule is this region there.

Now in the explore task, I'm going to be exploring this idea of finding regions like this a bit further.

So here we are, Yasmin is planning her garden and here's what she says, everything closer to CD than AC will be a flower bed.

Everything within three metres of A will be gravel.

The rest will be grass.

Now what shape will the gravel be and what shape will the flower bed be and what areas can you find? Now well, before you start this, I want you to draw a rectangle that is 10 centimetres by 4 centimetres and we're going to use those to represent 10 metres and 4 metres.

So we're going to say one centimetre in our scale is equal to one metre.

And once you've done that, give this task a go, okay? Pause the video now.

Okay, now that you've tried this, let's see what you should have done.

Well, we want regions that are closer to CD than AC.

So we want to be closer to CD than we are to AC.

So what do we do? Well we construct an angle bisector.

And you see? So anything in this region is closer to CD than it is to AC.

And then we are told all of that, so all of this will be a flower bed.

And then everything within 3 metres of A will be gravel.

We want to be the same distance from a single central point, the same distance from a single central point.

What do we have to construct? We have the same distance from a single central point.

What shape do you know that's the same distance from a single central point? And it is a, good, circle.

So if you open your circle to 3 centimetres, that's your circle.

If you open a compass to 3 centimetres, you can draw a circle around point A and everything inside that circle and in the rectangle, that will be gravel.

And if you notice, this makes a quarter of a circle 'cause that angle there is 90 degrees.

And the rest of the area so everything, so all of this, all of this will be grass.

So this is the picture that you should have come up with.

Now what areas can you find? Well, the flower bed, if you look at it is a right angle triangle and the gravel, well that is a quarter circle and in here, well, this shape for the grass is a compound shape, so it's a combination of many different shapes.

Now which areas can you find? Now if you have found some of those areas and you want to share your work, ask your parent or carer to share work on Twitter tagging @OakNational and #LearnwithOak.

Thank you very much for participating in today's lesson, I will see you again next time.