Draw a triangle 𝐴𝐵𝐶 in which 𝐴𝐵 equals three centimetres, 𝐵𝐶 equals four centimetres, and 𝐴𝐶 equals five centimetres. Bisect angle 𝐵 with a line that intersects line segment 𝐴𝐶 at the point 𝐷. Measure the length of line segment 𝐵𝐷.
To answer this question, we first begin by constructing a triangle 𝐴𝐵𝐶. Then, we’re going to construct an angle bisector to bisect angle 𝐵. In questions like this, we need to be as accurate as we can with our measurements and our drawings. Let’s have a look at some of tools we need to answer this question. The first kind of tool we need is this one. You may know it as a pair of compasses, a drawing compass, or simply a compass. But when we’re talking about a compass for drawing circles and arcs, we’re not talking about the tool that helps us navigate.
Let’s have a little look at our compass. It has a sharp pointed end which, if we were drawing a simple circle, would create the centre of the circle. At the top, there is a screw. So if the legs on the compass are too wobbly, then we need to tighten this screw. On the other side of our compass, we can put our pencil through. And we could make sure that the nut is tight so that the pencil doesn’t move up and down. When we’re using the compass, it’s best to try and use it with one hand holding on at the top rather than one hand on each leg of the compass. Let’s begin now by constructing our triangle. And we need a sharp pencil and a ruler as well as the compass.
In this triangle, we know that there’s going to be three corners labelled 𝐴, 𝐵, and 𝐶. And we have to draw three different lines. We can start by picking any one of those lines, but sometimes it’s helpful to pick the longest line. So, using our ruler, we can draw a line of exactly five centimetres long. And we label this as 𝐴𝐶. Now, we need to draw our line 𝐴𝐵, which is three centimetres, and a line 𝐵𝐶, which is four centimetres. Let’s imagine we drew two lines very accurately, a three-centimetre line and a four-centimetre line. But yet, they don’t meet. It could take a long time of trial and error to draw lines of the correct lengths that actually meet. So this is where our compass is used.
Let’s start with our line 𝐴𝐵 of three centimetres. We take our compass, and we set it to three centimetres. That means we take the pointed end and we put it at zero centimetres on our ruler. And we move the legs of our compass until the pointed end of our pencil is at three centimetres. Then, we lift our compass, and we put the pointed end at point 𝐴 in our line and create an arc with our pencil. Now, we know that every point on this arc is exactly three centimetres away from the point 𝐴. And next, to draw the line 𝐵𝐶, we need to set our compass to four centimetres. So we need to open up the distance on our compass so that there’s four centimetres between the sharp pointed end and the tip of our pencil. Then, we lift our compass and put the pointed end at 𝐶 and draw an arc. Here, we know that every point on the arc is exactly four centimetres away from point 𝐶.
Now, by looking at the arcs that we’ve drawn, we know that there is one point which is exactly three centimetres away from 𝐴 and four centimetres away from 𝐶. And that’s the point where the arcs intersect each other. We can then join these up to create our perfectly constructed accurate triangle 𝐴𝐵𝐶. We keep our construction lines in place. We don’t erase them. In the next part of the question, we need to bisect angle 𝐵. That means we need to cut angle be exactly in half. To do this, we’re going to use our compass. Since we’re bisecting angle 𝐵, we’re going to put the pointed end of our compass at point 𝐵. We’re going to draw an arc on line 𝐴𝐵 and an arc on line 𝐵𝐶. It doesn’t matter what the distance that our compass is set at, but it is helpful to have it more than halfway along the line. And we keep our compass set at the same size for each arc.
Now, we’re going to draw two more arcs. For the first point, we put the pointed end of our compass at the point where the arc has intersected with the line 𝐴𝐵. For our second arc, we’re going to start with the pointed end of our compass at the point where the arc intersects line 𝐵𝐶. And there should be a point of intersection of these two new arcs. We can draw a line through this point of intersection from point 𝐵. This line is the angle bisector. In this case, it has bisected angle 𝐵. Returning to the question then, we have bisected angle 𝐵 with a line that intersects 𝐴𝐶. This means that the point of intersection must be point 𝐷. Now, we need to measure the length of 𝐵𝐷. So using a ruler and measuring very carefully and accurately, we should find that the length of line segment 𝐵𝐷 is 2.4 centimetres.