Lesson video

Hello, I am Mr. Gratton.

Thank you all so much for joining me in this lesson on constructions and loci.

Make sure you have a pair of compasses and a ruler on hand for this lesson where we will use constructions to help us solve problems that involve a combination of loci.

Pause here to have a quick look at some important keywords that might come in helpful during this lesson.

First up, let's look at identifying and interpreting some loci problems in a given context.

Jacob here remembers that there are a lot of different Loci constructions.

Some that show a locus around a single point, some that show the locus of points around a line segment, and some that show the locus of points around a polygon or other shape.

However, in reality, not every space is completely empty or unobstructed.

Sometimes, different objects can just get in the way.

This makes our locus constructions look a little bit different.

For example, a robot vacuum cleaner can move up to nine metres away from its sensor.

What if the robot is placed in a room with the sensor near a wall? What would the locus of points the robot can reach actually look like? Well, there's nine metres.

Would the locus of points look like this passing through the wall? No, it wouldn't.

It would actually look like this.

The locus of points would be a segment of a circle, an arc for the part of the circle that's inside the room, and then a cord, a straight line segment at the wall of the room as the robotic vacuum cleaner obviously can travel across the edge of the wall, just not into it or beyond it.

Okay, let's test those real life limitations with some checks.

A goat is tied to a fence by a three metre long rope.

The fence is represented by each of these three vertical line segments.

There is a wall near the fence.

Pause here to identify the correct locus of points that the goat could walk to.

C is the correct answer.

A is the correct shape of a locus if there was no wall in the way, but C'S locus is adjusted for that wall.

B's shape is simply incorrect.

That is not the correct shape for a locus around a line segment at all.

Here's the next one.

This one's a little bit more tricky.

We have a motion sensor at point S that can detect motion of any object up to five metres away.

However, it can only detect motion in locations that it can see in a straight line.

Pause here to identify which of these shows the locus of locations that the motion sensor can detect.

B is the correct answer.

For B, it might be helpful to look at different radii from point S.

For example, this radius is obstructed by the car, so any point on that radius beyond the car cannot be detected by the sensor.

It can't see through the car to any point behind the car.

That's why C is incorrect.

C implies that the sensor can see points around the corner of that car, which is impossible for straight lines of vision.

We can use our construction techniques to identify different regions in a given context.

For example, this grid represents a rectangular garden filled with grass.

Towards the bottom of the garden, there is a plug socket at point P.

We've also got a shed on the top right of the garden as well.

Attached to that plug socket is a lawnmower with a seven metre long cable.

If we had no obstructions, then this would be the locus of points that the lawnmower can reach, but it makes sense that the shed cannot be mowed by the lawnmower so we exclude that part of the arc.

So let's shade in the region that the lawnmower cannot reach where there is still grass.

But before we shade in certain regions, let's try and identify where each and every region is, and what that region represents.

So, there are four different regions.

The shed is one region, and we have this main region.

This region includes points that are within seven metres of the plug.

Furthermore, we have these two other regions.

One at the top, and one very small region on the right.

Both of these regions have grass, but they are further than seven metres away from the plug.

So if we want to only shade in regions that the lawnmower cannot reach that still has grass, then we only care about the points that are further than seven metres away from the plug.

That means we only care about these two regions.

These two shaded regions are the two parts of the garden with grass that the lawnmower cannot reach.

Therefore, these are the two regions that have grass that cannot be cut.

Okay, here's a check for you.

Inside this room, we have a wifi router at point R.

We also have a TV and a cabinet.

Pause here to identify the regions that someone can stand in, but where the wifi signal cannot be picked up.

A, C, and F are the three regions that people can stand in where the wifi signal cannot reach.

The wifi signal can also not reach the TV and part of the cabinet, but because people can't stand there, they do not count.

We can also consider what different regions mean, and which regions satisfy certain relationships between two or more things such as multiple different locations on a map.

For example, on this map, we have two houses, A and B, and a radio tower at point T.

We want to find the locus of all points closer to A than B, closer to one house than the other, but these points must also still be within range of the radio tower's signal.

So, here are the correct constructions that break the map up into four different regions based on these two criteria.

At least one of these four regions satisfies the locus of points that we want.

Closer to A than B, but still within six kilometres of the radio tower.

However, what's the point in all of these regions if we do not actually understand what the constructions mean? Well, let's look into each arc and line segment one by one.

Pause here first to have a quick think or discussion about what each one might mean.

This line, this perpendicular bisector to AB shows the locus of all points equidistant to both points, A and B.

That means anything to the left of this perpendicular is closer to A, whilst anything to the right of the perpendicular is closer to B.

Next up, let's have a look at this arc, which bounds a segment.

The arc itself shows the locus of all points that are exactly six kilometres away from the radio tower.

Anything inside this segment is within the range of the signal of the tower.

Whilst anything outside of the segment is not within this range.

Okay, so let's bring all of that together.

Which region shows all points closer to A than B, but still within the range of the signal of that radio tower.

Well, this region shows the locus of all points closer to house A, and this smaller region both shows the locus of all points close to house A that are also within the signal of the radio tower.

This is the one region that we want.

Okay, so here's a map that shows Alex's house at point A and a river.

Alex's grandma wants to move to a house that is within seven kilometres of Alex's house, but at most, three kilometres away from the river.

Here is one construction that shows a locus of points.

Pause here to consider what might this arc locus represent.

It represents the locus of all points that are seven kilometres away from Alex's house.

The centre of this arc is point A, and its radius is seven kilometres.

Okay, for this same map, we have a river that terminates in an underground cavern.

Pause here to consider what this different locus represents.

This is a construction that shows the locus of all points around a line segment.

That line segment just so happens to be a river.

The maximum distance from the river or line segment to the locus is three kilometres.

Right, let's bring together both of those loci.

We now have three relevant regions.

Pause here to identify which region Alex's grandma can move to.

The correct region is Region B, but why isn't C a suitable region as well? Pause here to write down a sentence in the context of Alex's grandma explaining why region C is not suitable.

Region C is close enough to Alex's house, but it is far too far away from the river because it is more than three kilometres away from that river.

Great stuff, here's your first practise task.

For question one, this map shows a garden with a hose at point T and a shed.

Shade in the regions of the garden that the water hose cannot reach.

And for question two, we have a map that shows two different wifi routers at points A and B.

Each router has a signal distance of equal length.

Mark each of the three constructions, the two sets of arcs, and the one line segment that show each of the three different loci of points.

Pause here for these two questions.

Onto question three.

Here are three identical maps, each with different constructions drawn on them.

Shade in regions, and draw on a locus of points for each of these three criteria.

It might be helpful to measure the radius of each circle or arc.

Pause here for question three.

And finally, question four.

Here we have a complex map with a dog tied to a post, a picnic table, a wall, and an oak tree.

Pause here to shade in two different parts of the map.

One that shows the locus of all points that the dog can reach and still smell food from the picnic table, and the other region that shows where someone could sit to get shade without sitting on leaves from the oak tree.

Pause now for question four.

Lovely work, everyone.

Here are the answers.

Pause here to check if your shaded regions for question one and labelled construction for question two match these on screen.

For question three, pause here once more to check if your two shaded regions and locus of points match these on screen for three A, B, and C.

And finally, question four.

There is a tiny region that the dog can reach and also smell food from the table.

However, there is a much bigger region to the south of the wall that provides some shade, but is also not too close to the oak tree's leaves.

So we've looked at regions made from constructions that have already been given to us.

But now, let's construct our own locus regions and identify what each of those regions mean.

Let's have a look.

The language used in a context is incredibly important in helping us figure out which constructions we need to draw to show certain loci.

For example, Aisha wants to read in the park.

She needs to be within four metres of the lamppost to benefit from its light.

However, she also wants to be closer to the tree than the lamppost in order to take cover in case it rains.

Let's make some constructions to figure out where Aisha can sit in the park.

If a context says something needs to be within or beyond a certain distance away from a single point or single location, we usually need to construct a circle with a centre at that point.

So this circle shows the locus of all points exactly four metres away from the lamppost at point L.

Every point inside that circle shows the locus of all points less than four metres from the lamppost, whilst any point outside the circle is greater than four metres away from the lamppost.

But this context is talking about a comparison between two different distances, distances from both the tree and the lamppost.

In contexts where we have to compare two or more distances, we can construct a perpendicular bisector between those two points.

So for the line segment TL, we construct four arcs each with equal radius.

To give two points of intersection for our perpendicular bisector to pass through.

This perpendicular bisector shows the locus of all points that are equidistant to both the tree and the lamppost.

Importantly, any point to the left of the perpendicular is closer to the tree, and this is absolutely important because Aisha needs to be within four metres of the lamppost to benefit from its light, but also importantly, closer to the tree to take cover in case it rains.

We want a region that is both inside this circle, but also to the left of the perpendicular bisector.

This is the only region that Aisha can sit in.

Okay, here's another scenario for you.

Pause here to identify the construction Jun could use to figure out whether he is within 1,000 metres of either of the two supermarkets marked A or B.

We're looking at the distance from a single point, point J.

Therefore, a circle with centre at J will do this.

And here's the circle.

Since both points A and B are outside of the circle, both supermarkets are further than 1,000 metres away from Jun.

So since Jun is over 1000 metres away from both supermarkets, he now wants to figure out which supermarket he is closer to.

Pause here to identify the construction Jun could use to figure this out.

We can draw a perpendicular bisector between points A and B.

Anything to the left of the bisector will be closer to A, and anything to the right closer to B.

This is the line segment AB, and these are the construction arcs each with equal radius.

Therefore, this is the perpendicular bisector.

Jun is ever so slightly closer to supermarket A than B.

However, location is not the only thing that you need to consider.

It is also absolutely important to take the specific context you're dealing with into account as well when considering appropriate loci and regions.

In this example, we have a security camera at point S.

This camera is placed on the outside wall of a house where it can capture video footage up to 20 metres away from the house.

However, it can only capture video footage if that location is bright enough either from the sun, or artificial light at nighttime.

At night, there is a lamppost that shines bright enough to be captured, as well as any location up to 12 metres away from the lamppost.

There will be different loci of points that the security camera can capture between the daytime and the nighttime.

So let's start off with an unobstructed locus and then refine it by factoring in any of the locations on the map.

This arc is the locus of all points exactly 20 metres away from the security camera.

But we need to factor in that the camera cannot capture video footage from behind the van, such as with that point there.

If we draw a straight line segment from point S where the security camera is, to that point behind the van, the line segment must pass through the van.

Meaning that the camera cannot see that point.

To find the locus of all points that the security camera can actually capture footage from, we need to exclude areas by looking at the vertices of objects that are in the way such as that van.

So let's create rays from point S through vertices of the van like so.

One ray for each vertex of the van that the security camera can see.

If you are in any doubt, draw more rays than necessary from point S, and then see which rays are helpful.

This now dotted locus of points are points that the security camera cannot see because the van is in the way, even though those points are within the appropriate distance from point S.

Furthermore, the security camera can also not look behind the house because it cannot see beyond the walls it is attached to.

Therefore, we must also draw these two line segments that are extensions of this edge of the house wall.

So let's take into account all of this context.

These two parts of the rays that are close to point S are excluded from this locus, as they are definitely visible points on the security camera.

We only use this ray to figure out points further beyond that the van obstructs.

Inside this pretty oddly shaped region is the locus of all points that the security camera can capture footage from during the daytime.

However, during the nighttime, we also need to consider the lamppost as a light source.

Therefore, we need to consider the intersection between the daytime region we just looked at, and the locus of all points less than or equal to 12 metres from the lamppost.

Therefore, this even more oddly shaped region, this smaller region than the daytime region is the locus of all points that the security camera can capture nighttime footage of.

Right, onto this final situation.

Lucas at point L shines a torch that reaches up to 20 metres away.

The arc that we can see shows all points exactly 20 metres away from Lucas.

Pause here to identify the correct statements.

Absolutely none of these regions in the diagram correctly show where the torch will reach.

Pause here to think about or discuss why in the context of light coming from a torch and how it interacts with a wall.

The light from a torch cannot pass through walls or bend around walls.

So here are some rays that show the trajectory of light in specific directions.

Some of these rays are in directions the light can reach, and others in directions where the light cannot reach due to the walls.

Pause here to identify which of these rays will help identify the locus of points the torch can reach.

Only ray C will help us.

This is because this ray identifies a vertex of the wall that is currently unobstructed.

Both A and B are also rays to vertices.

However, these vertices have parts of the wall in the way first.

Meaning that the rays cannot reach that vertex.

Which of these four regions separated by this ray and the arc will the torch reach? Pause now to identify the correct one.

The light will only reach region D.

This is the region you would shade in to represent the locus of all points that Lucas's torch can reach.

Great stuff, everyone.

Onto the practise task.

For question one, here we have a dog tied to a pole at point D.

Shade in the region of points that the dog can reach, factoring in that the fence is in the way.

And for question two, identify and shade in the locus of all points that a bin can be placed if it needs to be within three metres of the bench, but more than five metres away from the tree.

Pause now for these two questions.

Onto question three.

There are two radio towers at point X and Y.

There are also seven houses marked A to G.

Classify each house regarding whether they can receive signal from each of the two radio towers.

Pause here for this question.

And finally, onto question four.

Shade in the region of all points that a statue can be placed when considering a lamppost and flower bed, and also shade in the region of all points that a motion detector can detect motion from.

Pause now for this last question.

Oh, amazing work on all of those constructions.

Pause here to compare your shaded regions for both questions one and two to those on screen.

For question three, there are three constructions that you should have drawn.

Two different circles or arcs, really.

One with a centre at point X, and the other at point Y.

Furthermore, we can use these two arcs to construct a perpendicular bisector to the line segment XY.

Each house now falls into one specific category regarding its distance from the two radio towers at X and Y.

Pause here to check your answers to these seven houses, versus those on screen.

And lastly, here's question four.

The statute can be placed in this fairly small region towards the top of the map.

Whilst the motion detector can detect motion in this quite large region that is obstructed by the storage building and the wall.

Pause now to compare your regions to the ones on screen.

Wow, great work, everyone, on all of those constructions, and then analysing their relevance to a given context.

In a lesson where we have looked at constructions that could help us solve various real world loci problems where the type of construction will change depending on the context that is given.

Also, different contexts will involve different regions being relevant.

We shade in those relevant regions to show their importance.

Keeping construction lines in is absolutely always necessary, and also they can be important in subsequent constructions.

Furthermore, constructions involving sight or light may look different if there are objects that are in the way.

Once again, amazing effort, everyone, in this lesson and in any other construction and loci lesson you might have done in the buildup to this one.

Thank you all so much for your hard work and your effort in everything that you do.

I've been Mr. Gratton, and you have all been absolutely spectacular students.

Take care all, and goodbye.