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)
`(7)/(8)a`

B)
`(5)/(8)a`

Explanation:

AB has length a and is divided by points P and Q into AP , PQ , and QB , such that AP = 2PQ = 2QB

A) Therefore, AP = 2QB

QB = AP/2

The midpoint of QB = QB/2 = (AP/2)/2 = AP/4

AP = 2PQ, Therefore PQ = AP/2

Since the length of AB = a

AB = AP + PQ + QB = a

AP + AP/2 + AP/2 = a

AP + AP = a

2AP = a

AP = a/2

The distance between point A and the midpoint of segment QB = AP + PQ + QB/2 = AP + AP/2 + AP/4 = 7/4(AP)

But AP = a/2

Therefore The distance between point A and the midpoint of segment QB = 7/4(a/2)=
`(7)/(8)a`

B)

the distance between the midpoints of segments AP and QB = AP/2 + PQ + QB/2 = AP/2 + AP/2 + AP/4 = 5/4(AP)

But AP = a/2

Therefore the distance between the midpoints of segments AP and QB = 5/4(AP) =
`(5)/(4) *(a)/(2)=(5)/(8)a`