Segment AB has length "a" and is divided by points P and Q into AP, PQ, and QB, such that AP = 2PQ = 2QB. a) Find the distance between point A and the midpoint of segment QB
b) Find the distance between the midpoints of segments AP and QB

a) 7a/8; b) 5a/8  

Step-by-step explanation:

Given:

AP = 2PQ = 2QB

Calculations:

1. A to the midpoint of QB

        a = AP + PQ + QB

If 2PQ = 2QB.

    PQ = QB and

    AP = 2PQ

∴      a = 2PQ + 2PQ = 4PQ

    PQ = a/4

    AP = 2PQ = a/2

Let M be the midpoint of QB.

AM = AP + PQ + QM

     = a/2 + a/4 + a/8

     = 7a/8

2. Midpoints of AP and QB

Let N be the midpoint of AP

NM = NP + PQ + QM

      = a/4 +a/4 + a/8 =

      = 5a/8

explanation:

previous steps have made pa≅ pb ≅ st ≅ su. setting the compass width to ab allows you to make tu the same length. then δstu ≅ δpab by the sss congruence postulate, and ∠p ≅ ∠s by the reason cpctc.