Jack wants to find the perpendicular bisector of the line segment. His method is described below. He measures half the length of the line segment and sets his pair of compasses to this length. He then draws two arcs from either end of the line segment. He draws a line between the two arcs. What is the error in his method?
So first, he’s measuring with his ruler the length of this line segment here. He’s then taking half of that distance and setting his compasses to that radius. He puts his compass point at one end and draws an arc, then puts his compass point at the other end and draws an arc. Finally, he draws a line between the two. Now, this does superficially look like a perpendicular bisector of the line segment. This part of the line is the same length as this part. So he has bisected the line. He’s cut it into two equal halves. But he did that by measuring it with his ruler rather than performing a construction. But what about making this line perpendicular to the original line segment?
There was nothing in what Jack did to guarantee that this angle here was 90 degrees. So let’s take a look at what he should’ve done in his construction. And then we can work out what he’s done wrong. Let’s ignore Jack’s constructions and go through how to construct the perpendicular bisector of this line segment.
First, we open up the compasses so that their radius is just over half the length of the line. We don’t need to measure this. We can do this visually. It doesn’t matter exactly what that length is, so long as it’s just over half the length. We then place the compass point carefully down at one end of the line. Then, we draw an arc making sure that each end of that arc comes back more than halfway. Then, taking great care to keep the radius of the compasses the same, we move the compass point to the other end of the line and drawn another arc. The second arc should intersect the first arc in two places, here and here. Then, we can carefully join those two points of intersection with a straight line.
Now, that process ensures that this side of the line segment here is equal in length to this half here. But it also guarantees that our perpendicular bisector is, in fact, perpendicular at 90 degrees to the original line. So how do we describe all that? Well, something like Jack should have set the radius of his compasses to greater than half the length of the line segment. This would’ve meant that his arcs created two points of intersection. He could’ve drawn a line through these points to create the perpendicular bisector. Okay, let’s go on and look at parts b and c of the question now.
Jemima wants to find any points that are equidistant from points 𝐴 and 𝐵. She decides that there is only one point, which she has highlighted in her answer below. What is wrong with her answer?
Well, equidistant means equally distant, the same distance away from points 𝐴 and 𝐵. Now, in her answer, this distance from 𝐴 is the same distance from 𝐵. So she certainly has found a point which is equidistant from points 𝐴 and 𝐵. The problem with her answer is that there are also an infinite number of points above and below the line which are equidistant from points 𝐴 and 𝐵. For example, this distance is the same as this distance, and this distance is the same as this distance, and so on. So if we constructed the perpendicular bisector of the line segment 𝐴𝐵 like we did in part a, then all the points on that line would be equidistant from 𝐴 and 𝐵. So what’s wrong with her answer is that Jemima has only identified the midpoint of the line segment 𝐴𝐵. But there are an infinite number of points equidistant from 𝐴 and 𝐵 on the perpendicular bisector of 𝐴𝐵. Okay, lastly then, let’s look at part c.
Triangle 𝐴𝐵𝐶 is shown in the diagram. Using a pair of compasses, construct the bisector of angle 𝐵𝐴𝐶. Show all construction arcs clearly.
Now, when a question tells you to use a pair of compasses to construct something, that basically means you’re not allowed to use a protractor to measure the angle. And when they say show all construction arcs clearly, they mean don’t rub out any working out that you do. Now, we just need to identify which is angle 𝐵𝐴𝐶. Then, we can get on with our construction. 𝐵𝐴𝐶 is a journey starting from 𝐵 going to 𝐴 and then going to 𝐶. So this angle here is angle 𝐵𝐴𝐶, the angle in the middle of that journey. So we’ve got to construct the line that goes through vertex 𝐴 and exactly cuts the angle 𝐵𝐴𝐶 into two equal-sized angles.
So we start by carefully placing the compass point exactly on vertex 𝐴 and opening up the compasses. The exact radius doesn’t matter. It needs to be large enough so that you can be accurate, but not too large so that it’s longer than side 𝐴𝐵 or side 𝐴𝐶. Then, carefully keeping the compass point on vertex 𝐴, we can draw an arc so that it intersects side 𝐴𝐶 and it also intersects side 𝐴𝐵 here. So the important point of this part of the construction was that we created two little line segments, this one here and this one here, that are the same length as each other.
Next, we’re gonna draw some more arcs that intersect each other, with our compass points at these two points, in turn. First, we’ll draw an arc here. Then, we’ll draw an arc here. And we can see that they intersect here. Now, we already said that these two lines are the same length. But because we kept our radius the same when we were drawing these other two little arcs, we also know that these two arcs are the same length as well. We’ve got a quadrilateral with all four sides the same length; it’s a rhombus. And we know that in rhombuses, the opposite angles are equal. So these two large angles are equal. And these two small angles are equal to each other as well.
So if we carefully draw a line through point 𝐴 and our little point of intersection up here, it exactly divides this angle into two equal parts. In other words, we’ve bisected angle 𝐵𝐴𝐶. And so, your finished construction should look something like this.