Line segment
AB
with the length ais divided by points P and Q into three line segments:
AP
,
PQ
, and
QB
, such that AP = 2PQ = 2QB. Find: (1) The distance between point A and the midpoint of segment
QB
; (2) The distance between the midpoints of the segments
AP
and
QB
.
To solve this problem, let's break it down into smaller steps:
1. Determine the value of each line segment:
Let AP = x, PQ = y, and QB = z. We know that AP = 2PQ = 2QB, so we have:
AP = 2y
PQ = y
QB = z
2. Express the length AB in terms of x, y, and z:
Since AB = AP + PQ + QB, we can substitute the known values:
AB = x + y + z
3. Find the values of x, y, and z:
Since AP = 2PQ = 2QB, we have:
x = 2y
y = y
z = y/2
4. Express the length AB only in terms of y:
Using the values of x, y, and z, we can rewrite AB as:
AB = 2y + y + y/2 = 5y/2
5. Find the distance between point A and the midpoint of segment QB:
The midpoint of segment QB is simply the average of its endpoints:
Midpoint of QB = (Q + B)/2 = (y + (y/2))/2 = (3y/2)/2 = 3y/4
The distance between point A and the midpoint of segment QB is the absolute difference between their coordinates:
Distance = |A - Midpoint of QB| = |0 - 3y/4| = 3y/4
6. Find the distance between the midpoints of segments AP and QB:
The midpoint of segment AP is (A + P)/2 = (0 + x)/2 = x/2
And we already found the midpoint of segment QB as 3y/4.
The distance between the midpoints of segments AP and QB is the absolute difference between their coordinates:
Distance = |Midpoint of AP - Midpoint of QB| = |x/2 - 3y/4| = |2y - 3y/4| = |8y/4 - 3y/4| = |5y/4| = 5y/4
So, the distance between point A and the midpoint of segment QB is 3y/4, and the distance between the midpoints of segments AP and QB is 5y/4.
Let the length of PQ = x
Then AP = QB = 2x
So, the midpoints of AP and QB divide AP into five equal segments.
Now you should be able to answer the questions easily.
I assume you have drawn the diagram ...