Loci are a set of points with the same property. Loci can be used to accurately construct lines and shapes. Bearings are three-figure angles measured clockwise from North.
A locus is a path formed by a point which moves according to a rule. The plural is loci.
Okay, we're going to be looking at loci and constructions today. Starting off with basic low chi and then finding the perpendicular from a point off the line. Perpendicular from a point on the line. The perpendicular bisector and bio dissecting an angle. These are all constructions that are done with compass and straight edge.
It says ruler in this diagram, but it's always a straight edge. An actual ruler with measurements is not allowed. It goes back to the ancient Greeks and Euclid's elements. And the ancient Greeks believe very strongly that a straight line and a circle were the two basic components of the universe. And so they wanted to construct everything out of just those two shapes.
So it's got a long and thousands and thousands of your history. The word loca, if you've never heard that before, it comes from the Latin locus meeting position or location. And it's the locus of points, the position where all the points cross. So, for example, if I have a straight line here and I wish to construct the locus of points that are exactly one metre from this line, you might be able to imagine that the locus of points one metre from this line is another parallel line, exactly one metre. Oh, wait, that's just what locus means, the locus of points, the collection of all of the points.
So we're going to be looking at that using a lovely app called Masspad. It's a free online software for you to use mathspad co UK. Highly recommended. Best one that I found. So we're going to jump in and use this Masspad co UK to demonstrate all the propositions today.
Okay, first things first. You have two basic loci or a locust. Singular. Loci is plural. You have two basic loci, one around a point and the other from a straight line.
So if we take the scale as being one metre, I want to construct the locus of points that are all one metre away from this point. In other words, all the points that are one metre away from this central point. How do I do this again? You might sort of figure out that it's a compass that's needed here because we're dealing with a circle with a radius. A circle is defined as the curve where all the points are equidistant from central point.
So we take our compass, we make sure it's unlocked and we match it to that one metre distance. We now lock it. This is what you would do in the classroom.
You then take your compass point onto the central point and you then rotate your compass around this central point. And so now you have constructed the locus of points exactly one metre from that central point. This circle is the locus. So that's you done and you have been successful. That's the locus.
If you wanted to construct, let's say, all the points that are equidistant from this line, AB that's this vertical line here on the left, all the points that are one metre from that line inside of this rectangle. Again, you match your compass, which we've already done. You take it there. In this case, it's not a compass, it would actually be a ruler, because we're not drawing around a point, it's a line. So we take a ruler, we do it to one metre, which in our scale here is about two centimetres on our scale.
But we do it to about there.
You get the idea. So you would be doing a straight line around about one metre distance. And so this would be your locus. If this is exactly one metre, this would be your locus here. It'd be a straight line up and down, parallel to this line.
And that's the basics of loci. It's either around a point or it's from a straight line, in which case it would be a straight line. You will get some more complicated questions, but those are the essential basics. So let's have a look at one question and see how we go with that.
Okay, in this question, a stadium is going to be built and between town A and B, the stadium must be no more than ten kilometres from town A and no more than eight kilometres from town B. And one centimetre represents two kilometres. We've got B and A, and we know that the stadium is going to be built somewhere in the region. That is within ten kilometres of A and within eight kilometres of B. So how do we do this?
Well, going back to our masspad co UK at again, we see here if this is town A and this is town B, we needed to construct a region, a locus around A, whereby all of the points were ten kilometres or less. And let's say we've done that with our ruler. We now rotate around here and we get a region around A where every point within this region is less than ten kilometres from A. I really should have joined A and B with a line here. And now we do the same thing with B.
We adjust our compass and we reduce it now to eight kilometres using the scale from the question. And now we do the same thing. We construct around B the locus of points around B. And in this intersection region here, these are all the points where you are within ten kilometres of A and within eight kilometres of B. So within this region and this region only, that is the location where the town could be built.
So this is just an example of a construction loci construction question that you might see in an exam.
Moving on from loci now to more particular Euclidean constructions, we are first of all going to construct the perpendicular that goes from this point that's off the line to the line. And essentially, these are all done the same way. You start off with your point and what you're going to construct is two circular ticks that are equidistant from each other on this line. And then all these examples. Basically, we're using this same principle.
And then that allows you to form some other ticks other way. And then you get your perpendicular. You'll see how that works? You must show your marking the marks from your circle. In the question, the phrase that is often used is no arcs.
That's a little arc there showing where it's gone through. No arcs, no marks. So we essentially are constructing a circle, but we don't have to draw the entire circle. We just have to draw a few arcs there to get the point across. Now, those two arcs, because we drawn a circle, they are equidistant from each other.
And we move here. And now we can extend this. It doesn't matter at this point. You couldn't extend it between those two. But now that we've done that, you can.
And we draw again two arcs here that cross from this point and then from the other second point that we drew that. And because of the properties of circles, because everything is symmetrical, we have just drawn where those two lines cross a point exactly underneath this point. And so when we join those, we have formed a perpendicular line that cuts this line exactly at right angles. And so we have constructed our perpendicular line from the point to this line.
Next up, we have a point on the line, and we're going to again construct the perpendicular from this point on the line. So we take our compass and the same idea applies. We are going to form two equidistant points by which we forgot to lock it. Well, let that be a lesson to you. In the real world, if you don't lock your compass, that can happen.
It goes in and out, changes its size, and then everything gets thrown off. Don't forget to lock your compass. So we'll draw this arc over here and flipping around to do the circle on the other side. This arc here. Now we have our two arcs either side.
And from this, the construction of the perpendicular is straightforward enough. So we will move this further out. Extend the size of it because you want your arc to be higher than the halfway point. Otherwise they're not going to cross. You'll see what I mean in a second.
So I form my arc from my circle there, and going to the other side, I do the same thing.
And where these two have crossed is directly above. We drop it down, and again, we have constructed the perpendicular to that line from a point that was on the line. Notice that we did essentially the same thing. We formed two ticks, one left, one right. And because of their symmetry, we then crossed two arcs above or below, which crossed exactly at the halfway point.
We're now going to construct the perpendicular to this line called the perpendicular bisector between these two points. Now, these two points already form our two left and right points from before, where there's a central middle point here. And we're going to get two arcs that cross above and below and join them up. So, once again, we draw our arcs above and below. Either way, we want them both on top and underneath.
And on the other side, there we go, there's one and rotating underneath, there is the other. And those two have now crossed. I'm joining up the points where those two cross has again formed the perpendicular line that splits these two points, this line, the distance between these two points in half. We have drawn the perpendicular bi sector by starting our compass point off at these two points and drawing a couple of arcs in the top and the bottom.
The last one you might see slightly different, although the same principle applies. It's bisecting an angle. In other words, this angle here is, let's say, 40 degrees. And we're going to cut that angle in half, making it 20 degrees and 20 degrees. We do almost the same thing.
We take this compass and we draw an arc from that central point, the vertex of the angle here. And this arc has now it's almost like this was the line from before. It sets up our left and right distance, from which we can draw an arc from the left point and an arc from the right point, very similar to the perpendicular bisector. And now, if we join up the central vertex to this point, this perfect symmetry here splits this angle in half. And so we have bisected the angle.
One of the great mysteries of ancient Greece was the ability to TRD an angle, split it in three, and no one was able to do that. And we now realise that with a compass and a straight edge, it's not possible.
Okay, and time for an example question. Using a ruler and a compass, construct the perpendicular bisector of the line AB. You must show all your construction lines. So you would have with you and the exam, the compass. You're not allowed to use a ruler for this question.
It's a compass construction. And so, moving back to the Masspad Cool UK app where I've drawn the line A B here, you just need to make sure that this compass distance is more than halfway to make sure the lines will cross. And then you can go ahead and construct your arc. Arc from the left and arc from the point on the right and similarly underneath.
There you go. And on the left.
And again, because of the symmetry that we've set up, this point above and below are perfectly above and below the halfway point and are by symmetry, cutting it exactly at right angles. And so we have now constructed the perpendicular bisector to the line AB. Easy.
And lastly, a question here for those of you who like a challenge, we have three points A, B and C on a map. One centimetre metre represents 100 metres. You'd have to use your ruler in an exam to adjust this properly. Point P is a new point that we're going to draw is 300 metres from A and it's Equidistant, which means the same distance from B and C. I'll give you a hint.
You are going to have to use construction methods that we have seen both loci and construction methods. Eucliding construction methods on this map show the possible positions of P. This is a region question similar to our town A and B. Much more difficult though. If you'd like to put your answers in the comments section or describe how it will be done, please feel free.
If you're not sure. If you have a question for me, myself or any other sherpa shooter, that's fine. My name is Adam. I myself regular shopping tutor, can answer that question for you. Good luck, guys.
This one's a toughie.
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