24. Loci What on earth is Loci? What we are going to do in this section Loci is all about tracing the paths of points as they move following certain rulesIt has many real-life applications, especially for architects and builders who want to make sure things go in the right place and they don’t run out of roomNote: If you are one of those people who doesn’t like the number and algebra bits of maths, then this could be the very topic for you!What we are going to do in this sectionInstead of going through how to do things like draw angle-bisectors, I am going to pick out a few of the classic type of Loci questions I have seen come up in exams in the past and take you through, step-by-step, how to do each one.NOTE: It is probably worth while reading through 8. Constructions before carrying on, as some of the skills you need are explained in greater detail there!
3Example 1 Scale:1cm = 2m Shed My pet penguin has been tied up by a 10 metre rope to the corner of the shed as shown below. Draw and shade the area which my penguin can moveSkills needed: drawing circles with compassScale:1cm = 2mShed
4Steps:1. Firstly, we need to sort out our scale – every 1cm square is equal to 2metres in real life – so the 10m rope our penguin is tied to is in fact… 5cms long!2. Now, we want to see how far our penguin can go in all directions. So, we must draw a circle with our compass (radius 5cm) and with the centre at the point on the shed where the penguin is tied.Watch Out! But that’s not the full story… because walls of the shed prevent the penguin from going quite as far upwards – he cannot walk through walls!He can go along the side of the shed to point B, which is 3cms away, and once he has reached this point, he can go another 2cms in any direction.3. So… we must now set our compass again and draw a circle with radius 2cm and centre at point B.4. We now have the area where the penguin can walk, so we can shade it in!
6Example 2 C D B E hedge hedge A Scale:1cm = 20m A farmer wants to lay a water pipe across his field so that it is equidistant from two hedges. He also wants to connect a sprinkler in the exact centre of the pipe, that waters the field for 40 metres in all directions.Skills needed: bisecting angles and bisecting linesShow the position of the pipe inside the field.(b) Mark the point of connection for the sprinkler.(c) Show the area of the field that is watered by the sprinkler.CDBEhedgehedgeAScale:1cm = 20m
7Steps: C D B E hedge hedge A Show the position of the pipe inside the field.CSteps:1. Firstly, we need to realise what the question is asking… the pipe must always be the same distance from line AB as line AE… well, the only way to do that is to bisect the angle at A!2. Place the pointy bit of your compass at A and mark a point on AE and AB3. Now place your pointy bit on each of these new points and draw two arcs in the centre of the shape4. Mark a new point where these two arcs cross5. Draw a line that starts at A and goes through this crossing point and voila!... There is your pipe!DBEhedgehedgeA
8Steps: C D B E hedge hedge A (b) Mark the point of connection for the sprinkler.Steps:1. Okay, so we have to find the exact centre of the pipe. Now it might be tempting to try to do it with your ruler… but that’s no fun, and more importantly, it’s not accurate! Instead, we must bisect the line2. Place the pointy bit of your compass at A draw an arc on the right and an arc on the left3. Place the pointy bit of the compass at the other end of the pipe and do the same.4. Mark two points where these arcs cross5. Draw a line through the two crossing points and where it hits the pipe is the exact centre!CDBEhedgehedgeA
9Steps: C D B E hedge hedge A Scale:1cm = 20m (c) Show the area of the field that is watered by the sprinkler.CSteps:1. First we must check our scale… 1cm = 20m, and we want to water 40m… so that is 2cm on our drawing!2. The water can travel 2cm in all directions, so we must draw a circle3. Place the pointy bit of the compass at the centre of the pipe and draw a circle with radius 2cm4. Shade in the circle and you are done!DBEhedgehedgeAScale:1cm = 20m