Find the length of line segment 𝐶𝐵.
Let’s have a look at this diagram which has two sets of parallel lines. Firstly, we have the line segment 𝐷𝐹 is parallel to the line segment 𝐴𝐸. We can consider these as part of the triangle 𝐴𝐸𝐶, and let’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, it’s the line 𝐷𝐹 which we can say is a line that’s parallel to one side. We can therefore say that the proportionality of the length on one side, 𝐶𝐹 over 𝐹𝐸, is equal to the proportionality of the length on the other side, 𝐶𝐷 over 𝐷𝐴.
We were asked to find the length of the line segment 𝐶𝐵. However, at the minute, we don’t have enough information to help us work out the length of the line segment 𝐸𝐵. So let’s see if we can use the remaining pair of parallel lines to help us. The line segment 𝐴𝐵 is parallel to the line segment 𝐷𝐸, and these are part of the larger triangle 𝐴𝐵𝐶. We can apply the side-splitter theorem again, this time stating that 𝐷𝐴 is parallel to 𝐴𝐵. And so, it must divide the other two sides 𝐴𝐶 and 𝐵𝐶 proportionally. We could therefore write another proportionality statement that 𝐶𝐸 over 𝐸𝐵 is equal to 𝐶𝐷 over 𝐷𝐴. However, this second statement doesn’t help us work out any missing lengths either. But let’s consider the two proportionality statements together.
We can observe that both 𝐶𝐹 over 𝐹𝐸 and 𝐶𝐸 over 𝐸𝐵 are equal to 𝐶𝐷 over 𝐷𝐴. And since these two proportionalities are equal to the same thing, then they must also be equal to each other. We can write that 𝐶𝐹 over 𝐹𝐸 is equal to 𝐶𝐸 over 𝐸𝐵. Using the diagram, we can actually write a value for three of these line segments. 𝐶𝐹 is equal to 15 centimeters, 𝐹𝐸 is six centimeters, and 𝐶𝐸 is the sum of these, so it’s 21 centimeters. We can then plug these values into the equation and solve to find the value of 𝐸𝐵. So 15 multiplied by 𝐸𝐵 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 𝐸𝐵 is 8.4, and we’re still working in centimeters.
Now that we’ve worked out that 𝐸𝐵 is 8.4 centimeters, we can calculate the required length of 𝐶𝐵. And so we add 15, 6, and 8.4, which gives us the answer that the length of the line segment 𝐶𝐵 is 29.4 centimeters.