Why is partitioning a directed line segment into a ratio of 1:3 not the same as finding the length of the directed line segment? The ratio given is part to whole, but fractions compare part to part. The ratio given is part to part. The total number of parts in the whole is 3 – 1 = 2. The ratio given is part to part. The total number of parts in the whole is 1 + 3 = 4. The ratio given is part to whole, but the associated fraction is .

Partitioning a directed line segment into a ratio of 1:3 means dividing the segment into two parts where one part is one-third the length of the other. To put it another way, if we have a line segment AB and we want to partition it at a point C such that the ratio of AC to CB is 1:3, then the length of AC is one-fourth of the total length of AB, and the length of CB is three-fourths of the total length of AB. This is because the two parts (AC and CB) must add up to the whole line segment AB, and their lengths are in a ratio of 1:3, leading to a total of 1 + 3 = 4 parts.

Therefore, the fraction that represents the length of the smaller segment AC relative to the whole line AB would be 1/(1+3), which simplifies to 1/4. Similarly, the fraction representing the larger segment CB relative to the whole line AB would be 3/(1+3), which simplifies to 3/4.

Finding the length of the directed line segment, on the other hand, is a different task. It requires measuring the entire distance from the starting point to the ending point of the line segment, which is unrelated to how the segment is divided internally. The ratio of 1:3 helps in partitioning the segment but does not provide the length of the directed line segment itself.