Question Video: Finding a Side Length in a Triangle Using the Relation between Parallel Lines Mathematics • 11th Grade

Find the length of line segment ๐ถ๐ต.


Video Transcript

Find the length of line segment ๐ถ๐ต.

Letโ€™s have a look at this diagram which has two sets of parallel lines. Firstly, we have the line segment ๐ท๐น is parallel to the line segment ๐ด๐ธ. We can consider these as part of the triangle ๐ด๐ธ๐ถ, and letโ€™s 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, itโ€™s the line ๐ท๐น which we can say is a line thatโ€™s parallel to one side. We can therefore say that the proportionality of the length on one side, ๐ถ๐น over ๐น๐ธ, is equal to the proportionality of the length on the other side, ๐ถ๐ท over ๐ท๐ด.

We were asked to find the length of the line segment ๐ถ๐ต. However, at the minute, we donโ€™t have enough information to help us work out the length of the line segment ๐ธ๐ต. So letโ€™s see if we can use the remaining pair of parallel lines to help us. The line segment ๐ด๐ต is parallel to the line segment ๐ท๐ธ, and these are part of the larger triangle ๐ด๐ต๐ถ. We can apply the side-splitter theorem again, this time stating that ๐ท๐ด is parallel to ๐ด๐ต. And so, it must divide the other two sides ๐ด๐ถ and ๐ต๐ถ proportionally. We could therefore write another proportionality statement that ๐ถ๐ธ over ๐ธ๐ต is equal to ๐ถ๐ท over ๐ท๐ด. However, this second statement doesnโ€™t help us work out any missing lengths either. But letโ€™s consider the two proportionality statements together.

We can observe that both ๐ถ๐น over ๐น๐ธ and ๐ถ๐ธ over ๐ธ๐ต are equal to ๐ถ๐ท over ๐ท๐ด. And since these two proportionalities are equal to the same thing, then they must also be equal to each other. We can write that ๐ถ๐น over ๐น๐ธ is equal to ๐ถ๐ธ over ๐ธ๐ต. Using the diagram, we can actually write a value for three of these line segments. ๐ถ๐น is equal to 15 centimeters, ๐น๐ธ is six centimeters, and ๐ถ๐ธ is the sum of these, so itโ€™s 21 centimeters. We can then plug these values into the equation and solve to find the value of ๐ธ๐ต. So 15 multiplied by ๐ธ๐ต 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 ๐ธ๐ต is 8.4, and weโ€™re still working in centimeters.

Now that weโ€™ve worked out that ๐ธ๐ต is 8.4 centimeters, we can calculate the required length of ๐ถ๐ต. And so we add 15, 6, and 8.4, which gives us the answer that the length of the line segment ๐ถ๐ต is 29.4 centimeters.

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