Question

How many different rays can be formed from three collinear points?

A visual expression would be useful not just using the equation 2(n-1) where n is the number of collinear points.

Answer 1

4 different rays that can be formed from 3 collinear points.

Explanation:

A ray is a part of a line that starts at a single point (called the endpoint) and extends infinitely in one direction.

A ray is named using its endpoint first, and then any other point on the ray, with an arrow on top, pointing in the direction of the ray. For example, the ray starting at point A and extending in the direction of point B is denoted as
`\overrightarrow{AB}`.

Collinear points are points that lie on the same straight line. Three or more points are said to be collinear if there exists a single straight line that passes through all of them.

Let the three collinear points be A, B and C (see attachment).

Each point can be the endpoint of a ray.

As point A if the left-most point, we can form one ray with point A as the endpoint. As this ray extends in the direction of points B and C, we can use either point B or point C as the directional point when naming the ray:

`\overrightarrow{AB}\;\;\left(\text{or}\;\;\overrightarrow{AC}\right)`

As point C if the right-most point, we can form one ray with point C as the endpoint. As this ray extends in the direction of points A and B, we can use either point A or point B as the directional point when naming the ray:

`\overrightarrow{CB}\;\;\left(\text{or}\;\;\overrightarrow{CA}\right)`

Finally, if we use point B as the endpoint of the ray, we can form two rays. As point B is between points A and C, we have one ray in the direction of point A, and the other ray in the direction of point C:

`\overrightarrow{BA}\;\;\text{and}\;\;\overrightarrow{BC}`

Therefore, there are 4 different rays that can be formed from 3 collinear points.