Answer: The distace between midpoints of AP and QB is [tex]\frac{a}{8}[/tex].
Step-by-step explanation: Points P and Q are between points A and B and the segment AB measures a, then:
AP + PQ + QB = a
According to the question, AP = 2 PQ = 2QB, so:
PQ = [tex]\frac{AP}{2}[/tex]
QB = [tex]\frac{AP}{2}[/tex]
Substituing:
AP + 2*([tex]\frac{AP}{2}[/tex]) = a
2AP = a
AP = [tex]\frac{a}{2}[/tex]
Since the distance is between midpoints of AP and QB:
2QB = AP
QB = [tex]\frac{AP}{2}[/tex]
QB = [tex]\frac{a}{2}*\frac{1}{2}[/tex]
QB = [tex]\frac{a}{4}[/tex]
MIdpoint is the point that divides the segment in half, so:
Midpoint of AP:
[tex]\frac{AP}{2} = \frac{a}{2}*\frac{1}{2}[/tex]
[tex]\frac{AP}{2} = \frac{a}{4}[/tex]
Midpoint of QB:
[tex]\frac{QB}{2} = \frac{a}{4}*\frac{1}{2}[/tex]
[tex]\frac{QB}{2} = \frac{a}{8}[/tex]
The distance is:
d = [tex]\frac{a}{4} - \frac{a}{8}[/tex]
d = [tex]\frac{a}{8}[/tex]
