Hello, el33t!

Let $\displaystyle A,B,C$ be three points in a plane such that: .$\displaystyle AB:BC \:=\: 3:5$

Which of the following can be the ratio $\displaystyle AB:AC$ ?

. . $\displaystyle \text{(I) }\;1\!:\!2 \qquad \text{(II) }\;1\!:\!3 \qquad \text{(III) }\;3\!:\!8 $

The answer choices are:

. . $\displaystyle \begin{array}{ccc}(A) & \text{I only} \\

(B) & \text{II only} \\ (C) & \text{III only} \\ (D)& \text{ I and III only} & \Leftarrow \\

(E) & \text{I, II and III} \end{array}$

$\displaystyle \text{(I)}\;1\!:\!2$ is possible.

The three points can be placed like this:

Code:

o C
* *
6 * *
* * 5
* *
* *
A o * * * o B
3

$\displaystyle \text{(III)}\:3\!:\!5 $ is possible.

The three points can be placed like this:

Code:

: - - - - - 8 - - - - - :
o---------o-------------o
A - 3 - B - - 5 - - C

But $\displaystyle \text{(II)}\:1\!:\!3$ is *not* possible.

Code:

: - - - - - - 9 - - - - - - : ?
o---------o-------------o
A - 3 - B - - 5 - - C