LOCUS POINTS & LOCI Studying the path of moving points and using them to make accurate drawings 12/01/2019

LOCI deals with the path drawn out by a moving point. The LOCUS of a point will show all the possible positions of that point as it follows a certain rule. × Lets have a look at an example × × Imagine a point on a piece of paper. × × Now draw another point which is 3cm away. Draw some more points which are all 3cm away from the original. × × × × × × Now lets put a few more on there. × Now lets shrink the crosses to small points and show all of the possible points which are 3cm away from the original. × × The green part shows all the possible positions of points which lie 3cm away from the original point. As you can see, they form a circle. WHAT IS THE RADIUS OF THIS CIRCLE? Answer: 3cm

LOCI deals with the path drawn out by a moving point. The LOCUS of a point will show all the possible positions of that point as it follows a certain rule. Lets have a look at another example Look at the line below and imagine all of the points which lie 2cm from it. Lets start to draw them in green. Which points have been missed out? Don’t forget the ones on the other side of the line too. The green part shows all the possible positions of points which lie 2cm away from the original line. As you can see, they form two parallel lines which lie 2cm at either side of the original line.

LOCI deals with the path drawn out by a moving point. The LOCUS of a point will show all the possible positions of that point as it follows a certain rule. Lets see what happens at the ends of a line. Lets draw all of the points which lie 2cm from this line. Whilst the line is straight, its just like the last example. A pair of parallel lines at either side of the original. What do you think happens at the ends of the line? Lets have a look to see if you were correct. The green part shows all the possible positions of points which lie 2cm away from the original line. The resulting shape looks similar to a running track

OK! Now its time for you to try one. Firstly draw an L shape with these dimensions. We are carefully going to draw all of the points which lie 2cm from this L shape. Before starting, try to imagine what it will look like when its finished. Lets start by drawing in those parallel lines, but be very careful on the interior corner. 4 Now we can deal with the ends of the lines. Use a compass set at 2cm for these Now lets deal with the corner. 3 The green part shows all the possible positions of points which lie 2cm away from the original L shape.

Now try these ones on your own. Draw each of the following shapes out using the dimensions shown and then mark on the lines which represent all the points which follow the given rule. Question 1: Mark on this diagram all the possible points which are 2cm from these lines. 1 3 Question 2: Mark on this diagram all the possible points which are 1cm outside of the triangle. 2 Question 3: Mark on this diagram all the possible points which are 1cm outside of this rhombus. Question 4: Using a pair of compasses, draw a circle of radius 2cm and then mark on all of the points which are 1.5cm from the circumference.

How well did you do? 2 1 3 3 3 3

Exam Style Question At a zoo, tigers are kept in a large enclosure. A tiger can reach through the bars of the enclosure and use its claws to grab anything up to a metre away. The zoo keepers want to erect a fence to make sure that visitors to the zoo do not stroll into the reach of the tigers. Draw out the enclosure from this diagram and show where they should erect their fence around this part of the cage. 3m Lets see how you did….. 2m 2m 3m 2m 2m 6m

× Using Loci Bob’s house lies exactly 200m from the Here are the points that are 200m from the river. Using Loci Bob’s house lies exactly 200m from the river and exactly 300m from the church. Use the position of the river and the church to find possible locations for Bob’s house. Church × Here are the points that are 300m from the church. So these are the only two possible positions of Bob’s house. River

A Practical Example A thief steals a helicopter and takes off from Bolton with 40 miles worth of fuel onboard. The police use a map to show where the thief could land the helicopter. Don’t forget that it is unlikely that he would land the helicopter in the sea

An Interesting Example Pick a point on the outside of a wheel and look at the path that it takes as the wheel rolls along the ground. The animation below traces out the locus of the point as the wheel turns.