Construct the locus of points that are the same distance from segments 𝐴𝐵 and 𝐴𝐶.
Now, when we’re constructing the locus of points, we’re simply tracing out a map of all the points that meet certain criteria. And in this case, we’re just mapping out all the points which are the same distance away from this line as they are from this line. Now this point, for example, there are lots of different lines that we could draw from that point that touch line 𝐴𝐵. But if we draw a perpendicular line to 𝐴𝐵 that goes through that point, that’s the shortest possible distance from the point to line 𝐴𝐵. And we could do the same from that point to line 𝐴𝐶.
Now, we can say that this point — let’s call it 𝑃 — is the same distance away from line segment 𝐴𝐵 as it is from line segment 𝐴𝐶. Simply, the shortest distance from that point to the line is the same in each case. Here’s another point 𝑄 we’ve done the same thing for and here’s another point, 𝑅. Now, if we trace a line through all of these points that are the same distance away from line segments 𝐴𝐵 and 𝐴𝐶, it would look something like this. Now, we’re gonna look at the construction: how do we actually construct that and work out where those points are.
So we start off with our diagram with line segments 𝐴𝐵 and 𝐴𝐶. Now, we’re gonna have to put our compass point at point 𝐴 in order to do this construction. So because I’m right-handed, I’ve actually turned the piece of paper round. So don’t be afraid to do that. It’s okay to turn the paper round to do your constructions.
So now, carefully place the compass point at point 𝐴 and open up the radius. Now the exact distance of that radius doesn’t matter. Generally speaking, the larger, the better because it’ll make more accurate constructions. in this case I’ve gone for about half the distance between 𝐴 and 𝐵. Now, extend that arc so that it cuts line segment 𝐴𝐶 and it cuts line segment 𝐴𝐵. And because we kept the same radius as we were drawing that arc, this distance here is the same as this distance here.
So we’re gonna take each of those points of intersection in turn and draw another arc, just out here somewhere. Now the exact radius that you use doesn’t really matter as long as you keep it the same for both of the next steps. It’s important that we get two arcs which intersect each other. So you may need to just adjust the radius up or down a little in order to create their point of intersection. So there’s our first arc. Now keeping that radius exactly the same, we draw another little arc from the point of intersection on line 𝐴𝐶.
Now because we can’t have radius the same for those two little arcs, this distance here is the same as this distance here. And don’t forget that this distance here was also the same as this distance here. And because of the symmetry of this, if we draw a line from 𝐴 through this point of intersection out here, then it will exactly cut angle 𝐵𝐴𝐶 in half and give us our locus of points that are same distance from segments 𝐴𝐵 and 𝐴𝐶.
So I just turn the piece of paper back round the right way. And that green line is our locus of points that are equidistant or the same distance from line segments 𝐴𝐵 and 𝐴𝐶. And because this is a construction question, don’t erase your construction marks. Those arcs that you drew to work out where the green line went are very important. So please don’t rub them out.