Question

Points P and Q belong to segment AB . If AB = a, AP = 2PQ = 2QB, find the distance: between the midpoints of segments AP and QB .

Answer 1

(7/8)a

Explanation:

First of all, the "question" (without a question mark) says "AB = a, AP = 2PQ = 2QB".

Segment AB is a, which is the total length of the segment with Points P and Q on/in it.

Since 2PG=2QB, it means that PG=QB.

In addition to this, AP=2PG and we can't simplify any constants, so we know that PG=QB and AP is double PG or double QB.

These are the only segments included in AB, so that means if we divide all the segments so that midpoints are included, we have (1/8)a for each part.

Since the question is telling us to find the distance between the midpoints of segments AP and QB, we have (8/8)a-(1/8)a, which is (7/8)a.

So, (7/8)a is the answer.

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Answer 2

`(a)/(8)`

Explanation:

AB = a

AP = 2PQ = 2QB

AB = AP + PQ + QB

= AP +
`(AP)/(2)` +
`(AP)/(2)`

AB = 2AP

AP =
`(a)/(2)` -----(1)

And we know QB =
`(AP)/(2)`

QB= AP/2 =
`(a)/(4)` ---------(2)

Now midpoint of AP =
`(a)/(2)` ×
`(1)/(2)` =
`(a)/(4)`

and midpoint of QB =
`(a)/(4)` ×
`(1)/(2)` =
`(a)/(8)`

distance between these midpoints =
`(a)/(4)` -
`(a)/(8)`

=
`(a)/(8)`