Find the lengths of line segment 𝐸𝐶 and line segment 𝐷𝐵.
If we look at this figure, we see that we have a larger quadrilateral 𝐴𝐵𝑌𝑋, and the larger quadrilateral is cut by the line segments 𝐸𝐹 and 𝐶𝐷. In addition to that, we see that we have four parallel line segments, two on the outsides of the quadrilateral and the two lines that cut the inside of the quadrilateral. And what we know is if three or more parallel lines intersect two transversals, then they cut off the transversals proportionally. This means that the transversal 𝐴𝑋 and the transversal 𝐵𝑌 are being cut proportionally by these four parallel lines.
And so we can say that 𝑋𝐸 over 𝑌𝐹 will be proportional to 𝐸𝐶 over 𝐹𝐷, which means we can say 14 over eight is going to be equal to the length of 𝐸𝐶 over 12. To solve for the length of 𝐸𝐶, we can do cross multiplication. 12 times 14 equals eight times 𝐸𝐶. 168 equals eight times 𝐸𝐶. To find 𝐸𝐶, we divide both sides of the equation by eight, and we get that 𝐸𝐶 is equal to 21. So we can say that the measure of line segment 𝐸𝐶 is 21 centimeters.
Now, we could set up another proportion exactly like this to find the measure of line segment 𝐷𝐵. However, because we know that these values are proportional, we should notice something else. 14 centimeters times two equals 28 centimeters. And this means that line segment 𝐷𝐵 will be equal to eight centimeters times two. And so we can say that the line segment 𝐷𝐵 is then equal to 16 centimeters. We’ve seen two different methods for solving proportional side lengths, and we found that 𝐸𝐶 equals 21 centimeters and 𝐷𝐵 equals 16 centimeters.