# How many rays can made from $4$ collinear points?

How many rays can made from $4$ collinear points?

The answer is $6$ (as floating around the internet) but I am not sure how is it possible, as far I know geometrically a ray is a line with one end point.

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We can think of a ray as being determined by two points: an endpoint and a second point that determines the direction. Let the four collinear points be $A$, $B$, $C$, and $D$, in that order.

• If $A$ is the endpoint of the ray, then all three choices for the other point, $\overrightarrow{AB}$, $\overrightarrow{AC}$, and $\overrightarrow{AD}$, are the same ray. So we have 1 ray with endpoint at $A$.
• If $B$ is the endpoint of the ray, then we have 2 possible rays, $\overrightarrow{BA}$ and $\overrightarrow{BC}=\overrightarrow{BD}$.
• If $C$ is the endpoint of the ray, then we have 2 possible rays, $\overrightarrow{CA}=\overrightarrow{CB}$ and $\overrightarrow{CD}$.
• If $D$ is the endpoint of the ray, then we have 1 possible ray: $\overrightarrow{DA}=\overrightarrow{DB}=\overrightarrow{DC}$

So there are $1+2+2+1=6$ possible distinct rays that we can name using those four collinear points.

edit Let me emphasize that I've made a jump in assuming that the intended question was "How many distinct rays can be named using pairs of points from the set of 4 collinear points?"

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So, for $n$ collinear points $2+(n-2)\times 2=2(n-1)$ – Quixotic Feb 4 '12 at 5:08
@MaX: Yes—the first and last points can only be used to name a single ray each, but every point in the middle can be used to name a ray in each of the two directions along the line. – Isaac Feb 4 '12 at 5:11

Let the points be $A,B,C,D$, in that order on some line segment. The rays are $AB$ extended, $BC$ extended, $CD$ extended, $BA$ extended, $CB$ extended, and $DC$ extended.

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