In triangle 𝐴𝐵𝐶, 𝐴𝐸 equals one centimeters, 𝐸𝐶 equals two centimeters, and 𝐴𝐵 equals six centimeters. Find the length of line segment 𝐴𝐹.
In this diagram involving the larger triangle 𝐴𝐵𝐶, we can also see that there are two parallel line segments, 𝐸𝐹 and 𝐶𝐵. We are also given the lengths of three different line segments. 𝐴𝐸 is one centimeter, 𝐸𝐶 is two centimeters, and 𝐴𝐵 is six centimeters. So we can add these onto the diagram. We are asked to find the length of the line segment 𝐴𝐹, which is part of this smaller triangle 𝐴𝐸𝐹. To do this, we’ll need to use the parallel lines to help us. These pair of parallel lines will allow us to apply 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. The line which is parallel to one side of the triangle is the line segment 𝐸𝐹, because it’s parallel to the line segment 𝐵𝐶. Because it intersects the other two sides, then these other two sides, 𝐴𝐶 and 𝐴𝐵, are divided proportionally by the line segment 𝐸𝐹. We could therefore write the proportionality statement that 𝐴𝐸 over 𝐴𝐶 is equal to 𝐴𝐹 over 𝐴𝐵. Substituting in the given length measurements will allow us to work out the unknown length of 𝐴𝐹.
Take care with the substitutions because the line segment 𝐴𝐶 will be the sum of the line segments 𝐴𝐸 of one centimeters and 𝐸𝐶 of two centimeters. We therefore have one-third is equal to 𝐴𝐹 over six. Cross multiplying, we have six is equal to three times 𝐴𝐹. And dividing both sides by three, we have two equals 𝐴𝐹. We can then give the answer that the length of the line segment 𝐴𝐹 is two centimeters.
Before we finish with this question, it’s worth noting that there are alternative proportionality statements that we could have written. We could have said, for example, that 𝐴𝐸 over 𝐴𝐹 is equal to 𝐸𝐶 over 𝐹𝐵. Substituting the values on the left side, 𝐴𝐸 is one centimeter and 𝐴𝐹 is the line segment that we wish to work out. On the right-hand side, we know that 𝐸𝐶 is two centimeters. The length of 𝐹𝐵 is one that we don’t know, but we can write it in terms of 𝐴𝐹 as six minus 𝐴𝐹. Writing it like this allows us to have just one unknown in our equation.
Cross multiplying, we have six minus 𝐴𝐹 is equal to two times 𝐴𝐹. Then we can add 𝐴𝐹 to both sides, which gives us six is equal to two times 𝐴𝐹 plus 𝐴𝐹. And we know that two 𝐴𝐹 plus 𝐴𝐹 is simply three 𝐴𝐹. Dividing both sides by three, we have two is equal to 𝐴𝐹. And so we have demonstrated how two different proportionality statements will allow us to find that the length of the line segment 𝐴𝐹 is two centimeters.