### Video Transcript

If the perimeter of triangle ๐ด๐ต๐ถ is equal to 9.7 centimeters, ๐ธ is the midpoint of the line segment ๐ด๐ถ, and the line segment ๐ท๐ธ is parallel to the line segment ๐ต๐ถ, find the length of the line segment ๐ท๐ธ.

So the first thing we can do is use a bit of information that weโve been given. And that is that the perimeter of the triangle ๐ด๐ต๐ถ is equal to 9.7 centimeters because what we can do is use this to help us calculate the length of the line segment ๐ต๐ถ. Because what we can say is that the line segment ๐ต๐ถ 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 weโre 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 ๐ต๐ถ 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 weโve got these lines marked on at ๐ด๐ธ and ๐ธ๐ถ, so our two line segments there. So therefore, what we can say is that line segment ๐ด๐ธ is equal to line segment ๐ธ๐ถ. So therefore, we can say that if we want to define line segment ๐ด๐ธ, 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 ๐ธ๐ถ is going to be equal to 1.45 centimeters.

So at this point, weโre thinking, โOkay, how does this help us to find the length of the line segment ๐ท๐ธ?โ 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, letโs look at the sides of our triangle.

Well, for our triangle, this tells us that ๐ด๐ธ over ๐ด๐ถ is equal to ๐ด๐ท over ๐ด๐ต, which is equal to ๐ท๐ธ over ๐ต๐ถ. So we can see here that Iโve put our fractions this way round, so Iโve 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 weโre trying to find the length of the line segment ๐ท๐ธ, we can say that ๐ด๐ธ over ๐ด๐ถ, and thatโs because we now know ๐ด๐ธ and we know ๐ด๐ถ, is equal to ๐ท๐ธ over ๐ต๐ถ. So we can say that 1.45 over 2.9 is equal to ๐ท๐ธ over 3.7.

So therefore, we can carry on our calculation over here. And weโre gonna have 0.5 is equal to ๐ท๐ธ over 3.7. So then, if we multiply both sides of the equation by 3.7, weโre gonna have 0.5 multiplied by 3.7 equals ๐ท๐ธ. So therefore, weโre gonna get ๐ท๐ธ is equal to 1.85.

So therefore, we can say that in answer to the question, the length of the line segment ๐ท๐ธ is equal to 1.85 centimeters. And we found that using the side splitter theorem.