In the diagram below, 𝐴𝐵 equals 10, 𝐵𝐶 equals 𝑥 plus one, 𝐶𝐷 equals 20, 𝐸𝐹 equals 10, and 𝐹𝐺 equals 10. Find the value of 𝑥 and the length of line segment 𝐺𝐻.
Let’s begin this question by filling on the given length measurements. We aren’t given any units of measurement, but these would all be length units. We are asked firstly to find the value of 𝑥 which appears as part of the length of line segment 𝐵𝐶. In order to work out the value of 𝑥 and indeed the length of the line segment 𝐺𝐻, we’ll need to use these four parallel lines. Lines 𝐴𝐸, 𝐵𝐹, 𝐶𝐺, and 𝐷𝐻 we’re all given are parallel. We also note that the other two lines 𝐴𝐷 and 𝐸𝐻 are both transversals. A transversal is a line that intersects two or more lines in the same plane at distinct points.
Knowing that we have these four parallel lines and two transversals means that we could apply Thale’s theorem, or the basic proportionality theorem. This theorem tells us that if three or more parallel lines intersect two transversals, then they cut off the transversals proportionally. And so we can say that the ratio of the line segment 𝐴𝐵 to that of 𝐵𝐶 must be equal to the ratio of the length of the line segment 𝐸𝐹 to that of the line segment 𝐹𝐺. And so we could write the proportion that 𝐴𝐵 over 𝐵𝐶 is equal to 𝐸𝐹 over 𝐹𝐺.
We can then substitute in the given length measurements and see if we can solve it for 𝑥. This gives us 10 over 𝑥 plus one is equal to 10 over 10. We can then simplify the right-hand side, since 10 over 10 is simply equal to one. Next, we can multiply both sides by 𝑥 plus one. So we have 10 is equal to one times 𝑥 plus one. And, of course, one times 𝑥 plus one is just 𝑥 plus one. Subtracting one from both sides, we have nine is equal to 𝑥. And so we’ve answered the first part of the question. We’ve worked out that 𝑥 is equal to nine.
For the second part of the question, we need to work out the length of the line segment 𝐺𝐻, which is here at the top part of the diagram. To work this out, we will apply the basic proportionality theorem once more. Now we know that the line segment of 𝐶𝐷 will be proportional to this line segment of 𝐺𝐻. We can write a proportionality statement involving these two line segments and either 𝐵𝐶 and 𝐹𝐺 or 𝐴𝐵 and 𝐸𝐹. Let’s use the line segments 𝐴𝐵 and 𝐸𝐹. We could therefore write the proportionality statement that 𝐴𝐵 over 𝐶𝐷 is equal to 𝐸𝐹 over 𝐺𝐻. When we substitute in the given lengths, we have 10 over 20 is equal to 10 over 𝐺𝐻.
We could then cross multiply and work out the length of 𝐺𝐻. However, we might also notice that the numerators on both sides of the equation are the same. They’re both 10. And so in order to be equal, then the denominators must be the same. We could therefore immediately work out that 𝐺𝐻 must be equal to 20.
It’s worth pointing out that even though in this question we have a number of pairs of congruent line segments, but this won’t always be the case. The proportionality theorem just tells us that these line segments have to be proportional. In this question, they were congruent as well. But now we can give the two answers: 𝑥 is equal to nine and 𝐺𝐻 is equal to 20 length units.