Video Transcript
Find the length of line segment ๐ถ๐ต.
Letโs have a look at this diagram which has two sets of parallel lines. Firstly, we have the line segment ๐ท๐น is parallel to the line segment ๐ด๐ธ. We can consider these as part of the triangle ๐ด๐ธ๐ถ, and letโs recall the side-splitter theorem. This theorem tells us that if a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides those sides proportionally. In this case, itโs the line ๐ท๐น which we can say is a line thatโs parallel to one side. We can therefore say that the proportionality of the length on one side, ๐ถ๐น over ๐น๐ธ, is equal to the proportionality of the length on the other side, ๐ถ๐ท over ๐ท๐ด.
We were asked to find the length of the line segment ๐ถ๐ต. However, at the minute, we donโt have enough information to help us work out the length of the line segment ๐ธ๐ต. So letโs see if we can use the remaining pair of parallel lines to help us. The line segment ๐ด๐ต is parallel to the line segment ๐ท๐ธ, and these are part of the larger triangle ๐ด๐ต๐ถ. We can apply the side-splitter theorem again, this time stating that ๐ท๐ด is parallel to ๐ด๐ต. And so, it must divide the other two sides ๐ด๐ถ and ๐ต๐ถ proportionally. We could therefore write another proportionality statement that ๐ถ๐ธ over ๐ธ๐ต is equal to ๐ถ๐ท over ๐ท๐ด. However, this second statement doesnโt help us work out any missing lengths either. But letโs consider the two proportionality statements together.
We can observe that both ๐ถ๐น over ๐น๐ธ and ๐ถ๐ธ over ๐ธ๐ต are equal to ๐ถ๐ท over ๐ท๐ด. And since these two proportionalities are equal to the same thing, then they must also be equal to each other. We can write that ๐ถ๐น over ๐น๐ธ is equal to ๐ถ๐ธ over ๐ธ๐ต. Using the diagram, we can actually write a value for three of these line segments. ๐ถ๐น is equal to 15 centimeters, ๐น๐ธ is six centimeters, and ๐ถ๐ธ is the sum of these, so itโs 21 centimeters. We can then plug these values into the equation and solve to find the value of ๐ธ๐ต. So 15 multiplied by ๐ธ๐ต is equal to 21 times six. This simplifies to 126 on the right-hand side. When we divide through by 15, we get that the length of ๐ธ๐ต is 8.4, and weโre still working in centimeters.
Now that weโve worked out that ๐ธ๐ต is 8.4 centimeters, we can calculate the required length of ๐ถ๐ต. And so we add 15, 6, and 8.4, which gives us the answer that the length of the line segment ๐ถ๐ต is 29.4 centimeters.
