Video Transcript
If the perimeter of triangle 𝐴𝐵𝐶 is equal to 9.7 centimeters, 𝐸 is the midpoint of the line segment 𝐴𝐶, and the line segment 𝐷𝐸 is parallel to the line segment 𝐵𝐶, find the length of the line segment 𝐷𝐸.
So the first thing we can do is use a bit of information that we’ve been given. And that is that the perimeter of the triangle 𝐴𝐵𝐶 is equal to 9.7 centimeters because what we can do is use this to help us calculate the length of the line segment 𝐵𝐶. Because what we can say is that the line segment 𝐵𝐶 is going to be equal to the value we get when we subtract the lengths of the two sides away from the perimeter. So what we’re gonna have is 9.7 minus 3.1 minus 2.9, which is equal to 3.7. So therefore, we can say that the line segment 𝐵𝐶 is 3.7 centimeters.
Okay, great. But what do we do next? Well, if we take a look at our diagram, we can see that we’ve got these lines marked on at 𝐴𝐸 and 𝐸𝐶, so our two line segments there. So therefore, what we can say is that line segment 𝐴𝐸 is equal to line segment 𝐸𝐶. So therefore, we can say that if we want to define line segment 𝐴𝐸, well, this is going to be equal to 2.9 divided by two, which is gonna be equal to 1.45 centimeters. So therefore, we can also say that the line segment 𝐸𝐶 is going to be equal to 1.45 centimeters.
So at this point, we’re thinking, “Okay, how does this help us to find the length of the line segment 𝐷𝐸?” Well, what we can start to look at now is something called the side splitter theorem. What the side splitter theorem tells us is that if a line is parallel to a side of a triangle and the line intersects the other two sides, then the line divides those sides proportionally. But what does this mean in practice? Well, let’s look at the sides of our triangle.
Well, for our triangle, this tells us that 𝐴𝐸 over 𝐴𝐶 is equal to 𝐴𝐷 over 𝐴𝐵, which is equal to 𝐷𝐸 over 𝐵𝐶. So we can see here that I’ve put our fractions this way round, so I’ve have put the shorter sides on the top and the longer sides on the bottom. It can actually work the other way around just as long as we keep this consistent. Okay, great. So how can we use this to help us solve our problem? Well, as we’re trying to find the length of the line segment 𝐷𝐸, we can say that 𝐴𝐸 over 𝐴𝐶, and that’s because we now know 𝐴𝐸 and we know 𝐴𝐶, is equal to 𝐷𝐸 over 𝐵𝐶. So we can say that 1.45 over 2.9 is equal to 𝐷𝐸 over 3.7.
So therefore, we can carry on our calculation over here. And we’re gonna have 0.5 is equal to 𝐷𝐸 over 3.7. So then, if we multiply both sides of the equation by 3.7, we’re gonna have 0.5 multiplied by 3.7 equals 𝐷𝐸. So therefore, we’re gonna get 𝐷𝐸 is equal to 1.85.
So therefore, we can say that in answer to the question, the length of the line segment 𝐷𝐸 is equal to 1.85 centimeters. And we found that using the side splitter theorem.
