Lesson video

Hello, I'm Mrs. Lashley and I'm looking forward to working with you as we go through this lesson.

I really hope you're looking forward to it and you're ready to try your best.

So, our learning outcome today is to be able to apply the rules for constructing angle and line bisectors to loci problems. On the screen, there's some keywords that I'll be using during the lesson.

You've learned them before, but you may wish to pause the video and read them again just that you feel familiar before we make a start.

So, our lesson on applying constructions in loci problems is gonna be split into two learning cycles.

The first learning cycle is called perpendicular bisectors in loci problems, whereas the second learning cycle is angle bisectors in loci problems. So let's make a start at looking at perpendicular bisectors in loci problems. So two buoys are 60 metres apart, and the red cross in a circle are the icons I'm using for the buoys.

A boat is travelling between the buoys, so there is no closer to one than the other, and a boat is going to be indicated by a triangle.

So, where might the boat be? Well, it could be here, couldn't it? Because then it would be exactly the same distance from each buoy, so it's no closer to one than the other because it's the same distance.

But is it only equidistant, which means equal distance from both when it's in this position? So is that the only place the boat could be so there is no closer to one than the other? Well, no.

Anywhere that it creates an isosceles triangle with the two buoys would be the same distance.

So let's think about what that means.

Well, that's because an isosceles triangle has two equal edges.

So if we think about the three points here being the three vertices of an isosceles triangle, then the buoys would be our base vertices and the boat would be the third vertex.

So for example here, if you think about that boat now being in that position, the two edges, the edge between the boat and the first buoy and the edge and the other buoy would be equal in length because it's an isosceles triangle.

But it could also be here because those two edges would be, they're different lengths to the previous isosceles, but they are equal to each other.

And we need to make sure that this boat is no closer to one than the other.

It could also be here.

The isosceles triangle is now flipped down, if you like.

So how can the location of all of these boats be described? Can you see what's starting to appear? Well, we could describe them as being in a line, and this is the locus of the boat where it is equidistant from both buoys.

It is no closer to one than the other.

As long as it stays along that purple line, it's always created an isosceles triangle, and therefore it's equidistant from both.

What construction will give this locus? So remember, our lesson is about applying constructions to loci problems. So here is the locus of points that the boat could be on, that the boat could take, but what construction would give us that line? So it would be a perpendicular bisector.

So it's a perpendicular bisector of an imaginary line segment between the two buoys.

So we've got two points and we want to stay equidistant to both, and so we can use a perpendicular bisector in order to find that line.

So here is a check.

Which of the following points are not equidistant from both cones A and B? So imagine you're on a football pitch and you've got two cones and you need to stay equidistant from both.

Which points, so a, b, c, d, e, f, g, and h are your choices, are not equidistant to both? Pause the video.

And when you're ready to check, press play.

So by eye, we can see that a, d, f, and h are not equidistant to both cones.

b, c, e, and g are in that line.

And that line is the perpendicular bisector.

So, let's have a look at different problem.

This is a plan view of a parkland area.

Some children are playing tag while staying closer to the oak tree, which is the O, than the ash tree, which is the A.

So they're playing tag in a parkland area and they're staying closer to the oak tree than the ash tree.

So where can the children be so that they are closer to the oak than the ash? And Izzy says, "There's so many places.

Where to start?" Where would you start? And Sophia says, "Well, how about the line where they would be equidistant?" How about the line where they would be the same distance from both trees? And Izzy says, "Yes, that's their boundary." So if they're being told they got to stay closer to the oak than the ash, there is a line, an imaginary line, but a line that indicates where they would be passing through, that would be their boundary, where they become closer to the ash than the oak.

Sofia says, "We need to construct the perpendicular bisector." So once again, we've got two points here and we want to find the boundary line which will be equidistant to both.

So we can set up our pair of compasses to construct our perpendicular bisector.

And so this lines here that we can see is the boundary.

So Sophia says, "If any of the children are on this locus of points, then they're the same distance from each tree." "So the region that they can be on is the left of that line," Izzy said.

Because then they would be closer to the oak than the ash.

So here's a check for you.

Point A is the location of one hospital, and point B is the location of another hospital in a part of a map.

Describe the region that is shown here.

So I'm talking about the shaded grey region.

So pause the video.

And when you're ready to check your description, press play.

So the region that's closer to the hospital B than the hospital A.

Once again, the perpendicular bisector is gonna construct the locus of points that is equidistant from both hospitals.

So if we've shaded the region on the right hand side, then they are closer to hospital B than they are to hospital A.

So up to the first task of the lesson.

So question one, a robot is programmed to stay the same distance from two sensors at all times by constructing a perpendicular bisector show the locus of points that the robot can take.

So the two dots that are there are the two sensors.

So you need to construct a perpendicular bisector to show the locus of points.

Pause the video.

When you're ready for the next question, press play.

So on question two, you need to identify the correct region on each diagram.

So the description is underneath each one and you need to shade in the correct region.

So pause the video.

And when you finish doing those three parts, then we'll move to question three.

So on question three, this is a plan of a garden.

A bench is going to be installed.

It needs to be closer to corner A than corner B and further from corner A than corner D.

Identify the region that the bench will be located.

So pause the video.

Make sure you are constructing rather than just guessing where things would be.

You need to construct appropriate bisectors to locate that region.

Press pause, and then when you are ready for the answers to task A, press play.

So for the robot question, we can draw a sort of a line segment, that's an imaginary line segment, that connects the two sensors.

And then we need to do the perpendicular bisector.

The locus of points that I've drawn could be extended further.

On question two, you needed to shade the appropriate part of the diagram.

So I've used a light grey so you can see hopefully.

So the region that is closer to B than to A.

So the perpendicular bisector was drawn, and so that shows the locus of points that is equidistant, the same distance.

So if I need to be closer to B, then I'll be on the right hand side.

For part B, the region that is further from A than B, but that's just another way of saying the same thing.

If it's further from A, then it's closer to B.

So it's the same region as part A.

And for part C, the region that is equidistant from A and B is actually the locus is at the perpendicular bisector.

So you needed to just indicate it was that perpendicular bisector.

And for question three, identify region the bench could be located.

Would it be in the top right quarter? And that's because you needed to do two perpendicular bisectors and closer to corner A than corner B.

So they were two points, so you needed to do a perpendicular bisector.

And then further from corner A than corner D, so you need to do a perpendicular bisector.

You can see that I've not done the perpendicular bisector in the usual way.

I've used two isosceles triangles and then drawn the line of symmetry, if you like.

If you did the perpendicular bisectors in the more standard way and your construction marks are outside of the rectangle, that's absolutely fine.

This is just highlighting that there is an alternative way to constructing a perpendicular bisector if you're limited in space.

And that is making use of constructing two isosceles triangles with the same base edge and then connecting the points of intersection.

So we're now up to the second learning cycle, which is looking at angle bisectors in loci problems. So we've got a scenario here where two new houses are built so that there is land between them as shown.

The building company needs to instal the fence to separate the gardens.

The fence needs to be exactly halfway between both houses.

So you can see the plan view here of two houses.

So you've got house 1 and house 2.

The corners meet, and then the green space is the area that's garden, but they need to put in the fence so that it separates into two gardens, one for each house.

And that fence needs to be exactly halfway between both houses.

The first fence post will go here.

Could the next post be here? So just have a think about that.

So no, we would describe that to be closer to house 1 than house 2.

What about here? No, this time it's closer to house 2 than house 1.

Here? So yes, this is the same distance from both.

If you use to measure perpendicular to the edge of the house in both directions, it'd be the same distance.

What about this one? So yeah, that would be the same distance.

How about this one? Yeah, same distance.

So how can you describe the post locations? Remember we're trying to build the fence line.

So it's in a line that bisects the angle between the two house walls.

So if we think about the house walls creating an angle, there are two line segments that meet at a point at a vertex.

So there are our legs of our angle.

Then the line of fence posts needs to be exactly halfway and to it bisects the angle.

So looking at the two walls as the angle, the line that bisects the angle, which shows the locus of points that are equidistant from both legs, is the angle bisector.

So if we may take away the context and just leave the parts there, we can see that we could construct that using our angle bisector.

And so an angle bisector will always construct the locus of points that are equidistant from two line segments that meet at a point.

So here is a check.

Which of these points is equidistant from AB and BC? Pause the video.

And when you're ready to check, press play.

B and C are equidistant from both line segments.

The point that's marked A is definitely closer to AB than it is to BC.

And the point D is closer to BC than AB.

So here we have another scenario.

So if a fox stays closer to wall EF than wall FG, then it will not set off the security light.

So there's a security light that will be, the sensor will be set off if the fox doesn't stay closer to the wall EF than FG.

So what region can that fox roam without setting off the light? So Lucas says, "The two walls create an angle." So we can see E, F, G is our angle.

"An angle bisector will show the line of points which are neither closer to EF or FG." So the way Lucas has said that, an alternative way would be the line of points that are equidistant to both EF and FG.

So the region is above the locus of points that are equidistant that the fox can roam because then it is staying closer to the wall EF than the wall FG.

So here's a check for you.

Here is a plan of a garden where AD, CD, and BC are fences and AB is the house.

Describe the shaded region.

So you've got some construction marks on there as well.

So that might help you with your description.

Describe the shaded region in the context of the problem.

Pause the video, and then when you're ready to check, press play.

So it's closer to AD than AB and closer to AD than CD.

Instead of saying AB, you could have said the house.

So closer to AD than the house and closer to AD than CD, and AD, CD and BC are fences.

So we're up to the last task of the lesson.

Task B, question one.

By constructing an angle bisector, show the locus of points that are equidistant from AB and BC.

So you've got an angle there, A, B, C, and you need to construct the angle bisector such that you can show the locus of points that are equidistant from both legs of the angle.

Pause the video.

When you're ready for question two, press play.

So question two, shade the appropriate regions for each description.

So you've got three parts with a description underneath and you need to shade the appropriate region.

Press pause, and when you're ready for the final question, press play.

So here is question three.

A tree is to be planted in the garden ABCD.

It needs to be further from AB than AD.

Shade the region where the tree could be planted.

So pause the video.

Think about which bisector you need to use in this problem.

And when you press play, we're gonna go through the answers to task B.

So task B, question one.

You needed to construct an angle bisector to show the locus of points that are equidistant from AB and BC.

So the vertex of the angle was B, so that's where you would've put your pair of compasses to start with.

You then needed to construct the angle bisector.

So anywhere along that angle bisector is the same distance from both line segments.

On question two, you needed to shade the appropriate regions.

So there was two angle bisectors on each of the given diagrams. There was the angle bisector of ADC, and the angle bisector of ABC.

So on part A, you needed to shade the appropriate region, which was closer to CD than to AD.

So that was using one angle bisector.

So we needed to find the angle bisector, gave you the locus of points that is equidistant, the boundary if you like.

And then we need to think about where was closer to CD, and that would be on the right hand side of that locus of points.

For part B, you needed to be closer to CD than AD and closer to AB than BC.

So this is where you're gonna make use of both of the angle bisectors that were drawn.

So closer to CD than AD is where you shaded on part A, so to the right of the angle bisector, closer to AB than BC would be on the left of the second angle bisector.

So if it needs to satisfy both of those conditions, then it is only that sort of middle slice that would need to be shaded.

And lastly, on part C, closer to AB than BC and closer to AD than CD, so closer to AB than BC.

So if we think about the angle bisector for ABC, if it's closer to AB, then it's on the left hand side of that.

However, it also needs to be closer to AD than CD, which is used in the angle bisector of ADC.

So which parts satisfies both of them is only the top left because that is closer to AB than BC and it is also closer to AD than CD.

Finally, question three, the tree that's to be planted needs to be further from AB than AD and shade the region where it could go.

So we needed to think about AB as a line segment, AD as a line segment.

And so they're meeting at a vertex, which is A, so it's an angle.

So we're gonna use the angle bisector to find the locus of points that are equidistant and then consider which side is further from AB than AD or closer to AD than AB.

And so this is the region that the tree could be planted.

So to summarise today's lesson on applying constructions to loci problems, we can say that a perpendicular bisector can be used to construct the locus of points that are equidistant from two points.

And we saw that in a variety of different scenarios.

An angle bisector can be used to construct the locus of points that are equidistant from two connected line segments.

So if you can see an angle, because there's two line segments that might be the wall, might be a fence, but if they meet at a vertex, at a point, then we can do an angle bisector.

So if the region is closer to one point or a one line segment than the other, then the locus of equidistant points is necessary to mark the boundary.

So we start by constructing the boundary, the equidistant line, and then we consider which region we might need, whether it's closer or further away.

Really well done today and I look forward to working with you again in the future.