When we want to partition a directed line segment into a given ratio, it's essential to understand the difference between a part-to-part ratio and a part-to-whole ratio.
In this case, the ratio given is [tex]\(1:3\)[/tex]. This means that the line segment is to be divided into two parts where one part is 1 unit long, and the other part is 3 units long.
When we say the ratio is [tex]\(1:3\)[/tex], we are using a part-to-part comparison. To determine the total number of equal parts into which the line segment is divided, we add the parts of the ratio together. So, the total number of parts in the whole is:
[tex]\[ 1 + 3 = 4. \][/tex]
This total indicates that the entire directed line segment can be thought of as being divided into 4 equal parts.
If we were asked to find [tex]\(\frac{1}{3}\)[/tex] of the length of the directed line segment, this would be a fraction comparison of part-to-whole, not part-to-part. Specifically, finding [tex]\(\frac{1}{3}\)[/tex] of the segment is not the same as partitioning it into a [tex]\(1:3\)[/tex] ratio because the division of the line segment for a [tex]\(1:3\)[/tex] ratio means splitting it into 4 equal parts and comparing one part (1 part) to the whole (4 parts total).
Therefore, partitioning a directed line segment into the ratio [tex]\(1:3\)[/tex] corresponds to dividing the segment into 4 parts, where one part makes up [tex]\(\frac{1}{4}\)[/tex] or 25% of the whole length, and the other part is [tex]\(\frac{3}{4}\)[/tex] or 75%. This is not the same as taking [tex]\(\frac{1}{3}\)[/tex] or 33.33% of the whole length, which would be part-to-whole rather than part-to-part.
Hence, the correct detailed reasoning and solution are that the total number of parts in the whole is [tex]\(1 + 3 = 4\)[/tex].
