Final answer:
The question involves finding the sum of the possible lengths of segment PT on a line with specified midpoints and lengths. By considering the relationships between the points and segments, the possible lengths of PT are determined to be 12 and 14 units, summing up to 26 units.
Step-by-step explanation:
The question deals with a line segment divided into parts with given midpoints and lengths. We have points P, Q, R, S, and T in a line where Q is the midpoint of PR, S is the midpoint of QR, PR is 12 units and ST is 2 units. First, we determine QR's length since S is its midpoint. QR is the second half of PR, so it must also be 6 units (half of 12 units). As S is the midpoint of QR, QS and SR are 3 units each (half of 6 units).
Because ST is 2 units and S is to the left of T, the total length of the line from P to T (PT) can vary depending on where T is positioned relative to R. One possible scenario is when T is immediately to the right of S, giving PT = PR + ST = 12 + 2 = 14 units. Another scenario is when T coincides with R, making PT equal to PR, which is 12 units. Thus, the sum of the possible lengths of segment PT is 12 units (when T is at R) plus 14 units (when T is immediately to the right of S), which equals 26 units.