Question

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

Answer 1

a) 7a/8; b) 5a/8

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