Final answer:
To solve the problem, we calculated the length of segment PQ and found it equals a/5. Given that AP and QB are equal, the distance between their midpoints is the same as the length of PQ, which is a/5.
Step-by-step explanation:
To find the distance between the midpoints of segments AP and QB, we must first express AP and QB in terms of the given distances. Since AP is twice the length of PQ and PQ is half the length of QB, let's denote the length of PQ as x. Therefore, AP = 2x and QB = 2x.
Because AP is part of segment AB and AB = a, AP + PQ + QB = a. Substituting the expressions for AP and QB, we get 2x + x + 2x = a. Simplifying gives 5x = a, so x = a/5. Thus, AP = 2x = 2a/5 and QB = 2x = 2a/5.
To find the midpoint of AP, we take half of its length, which is (2a/5)/2 = a/5. Similarly, the midpoint of QB is also a/5 from point Q. Since AP and QB are equal in length and PQ is between them, the distance between the midpoints of AP and QB is simply the length of segment PQ, which is a/5.
Therefore, the distance between the midpoints of AP and QB is a/5.