Video Transcript
What is the total distance covered by someone who walks along the lines shown in the diagram, not walking on any line more than once?
Okay, so this is a question about distance. We’re presented with a diagram, and we’re told that we have someone who walks along the lines shown in this diagram. We are also told they are not walking on any line more than once. So with that information, let’s start by considering what that person’s journey looks like. The lines in our diagram form a triangle, and let’s imagine that our person begins at the bottom-left corner of this triangle. We can use the information given to us in the question to work out this person’s path.
There are two relevant pieces of information. The first is that they walk along the lines shown in the diagram, which we can take to mean that they walk along all of the lines. The second piece of information is that they don’t walk along any line more than once. Both these facts taken together tell us that the person walks along each line exactly once. With this knowledge, we can draw the person’s path onto the diagram. First, the person walks along one of the sides of the triangle. Then they walk along the next side. And last of all, they walk along the third and final side of the triangle, ending up back where they started.
It’s worth pointing out that this is not the only path they could have taken. Our person had two options for the first line that they chose to walk along, and they could have walked along the three lines in the opposite direction. It turns out that it doesn’t actually matter which of these two paths they take. The total distance covered is the same in either case. To see why this is, let’s recall our definition of distance. Distance is defined as the length of the path between two positions. In our case, the start and end positions are the same, this bottom-left corner of the triangle. The distance covered by the person is the length of the path they take along the three lines in the diagram to get back to this point.
For now, let’s imagine that our person follows the first path we drew: first, walking along the line we’ve labeled one, then the line we’ve labeled two, and, finally, the line labeled three. The distance covered is given by the length of this path which is equal to the length of line one plus the length of line two plus the length of line three.
Now let’s consider what would happen if the person walks along the second path that we drew. Well, in this case, they would’ve first walked along the line we’ve marked three, then the line we’ve marked two, and finally the line marked one. And, as before, the distance covered is given by the length of the path that they walk, which in this case is given by the length of the first line they walk along, line three, plus the length of the second line they walk along, line two, plus the length of the third line they walk along, line one.
Now when we’re adding together numbers, the order of the addition doesn’t matter. If we calculate the sum 𝑎 plus 𝑏, we’ll get the same result as if we calculate the sum 𝑏 plus 𝑎. So, we should see that these two expressions must give the same result. In other words, whichever of the two paths the person chooses to take, the distance covered will be the same. So let’s calculate this distance. The diagram tells us the length of each of the lines. All three lines have a length of six meters, and so the total distance covered is given by the length of line one, which is six meters, plus the length of line two, which is also six meters, plus the length of line three, which again is six meters. If we calculate this sum, we get a result of 18 meters.
And so we have our answer to the question that the total distance covered by someone who walks along each of the lines of the diagram not walking on any line more than once is given by 18 meters.
