Question 1089450: Five points A,B,C, D and e lie on a line. Point b is the midpoint of AC, and point D is the midpoint of BC. If AC is 12 and DE is 2, what's the sum of the possible lengths of segment AE?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! AB=BC=(1/2)AC because B is the midpoint of AC,
So, if AC=12, then AB=BC=6.
BD=DC=(1/2)BC because D is the midpoint of BC,
and because BC=6, BD=DC=3.
So, the line, with points A, B, C, and D,
and the distances between them, looks like the drawing below (or its mirror image).
 .
We do not know exactly where point E is located on that line,
but we know that DE=2.
It could be like the drawing below (or its mirror image).
 .
In that case,
AE=AB+BD-DE=6+3-2=7 .
Otherwise, it would be like the drawing below (or its mirror image).
 .
In that case,
AE=AB+BD+DE=6+3+2=11 .
The sum of the possible lengths of segment AE is
 .
If we did not know the distance AC, but we knew that DE
AE=AB+BD-DE or AE'=AB+BD-DE ,
so AE+AE'=AB+BD-DE+AB+BD+DE=AB+BD+AB+BD=AD+AD=2AD .
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