Number Properties (3rd Ed), Ch.8, Page 135 #8

[kath.alessandra]

Quick question for the following problem:

A,B, and C all lie on a line. D is the midpoint of AB and E is the midpoint of BC. AB = 4 and BC = 10. Which of the following could be the length of AE? A) 1 B) 2 C) 3 D)4 E) 5

Based on the answer explanations in the book there are only two possible arrangements for the points on the number line.
Why only two?
Why could the line not be arranged as follows, with the points from left to right: B - C - A OR C - A - B ???

I don't understand that based on the information in the question that we can conclude that there are only two possible arrangement for the points on the number line.
Please help! Thanks in advance!

[apratim2002]

B - C - A is not possible because in that arrangement, BC < AB. But according to the problem BC > AB (AB = 4 and BC=10). As for C - A -B, it is essentially the same as B - A - C already discussed in the solution on page 137. Basically what matters is the relative positions of the points. In general do not consider mirror images of arrangements which you have already considered. That would not produce any different result. So for an arrangement with 3 point A, B and C, there can be only 3 "different" arrangements :
1. With A in the middle
2. With B in the middle
3. With C in the middle
In the specific example, 1 and 2 is already discussed on page 137. And I explained why the 3rd one (C in the middle) is not possible. Exchanging the extreme points (points other than the one in the middle) will not produce any "different" result.

[tommywallach]

Hey Guys,

Apratim nailed the explanation here. Well done!

Let me know if you have any follow-up questions!

-t