How many segments are formed by 10 collinear points?

Question

How many segments are formed by 10 collinear points? Geometry help, I can't seem to find the answer please help.

Answer 1

If the segments are finite in length there are _nC₂ for n collinear points.  If you count the infinite segments, each point defines two, and so you must add 2n.  For n=10 the answers are 45 and 65 respectively.

Answer 2

Hi Andrea,

 

I've drawn a sample of a segment with 10 collinear points.

 

As you can see they all lie on the same line.

 

.___.___.___.___.___.___.___.___.___.

A    B    C     D    E     F     G    H     I     J

 

 

Now take a look at all of the possible line segments that you could make beginning with point A.

 

Segments AB, AC, AD, AE, AF, AG, AH, AI and AJ (9 segments)

 

Now take a look at point B.

 

You already counted BA (that's AB)

 

You have segments BC, BD, BE, BF, BG, BH, BI, BJ (8 segments, one less than 9)

 

In fact as you keep track from C to I, you will always have one less segment.

 

A (9 segments)

B (8 segments)

C (7 segments)

D (6 segments)

E (5 segments)

F (4 segments)

G (3 segments)

H (2 segments)

I (1 segment)

 

Total = 40 different segments in all

Answer 3

Hi Andrea--

 

co-linear just means they live on the same line.

 

So draw a line, with arrows on either end to show it goes on forever.

 

Now, place points on that line, as if you were making a number line--but don't worry about the numbers.

 

Just make sure you have enough room for 10 points.

 

Now, start at the first point you drew, and count the one segment between that point and the next point; then the next segment, etc., till you get to your last point, and stop there.

 

How many segments did you count? (If it's not 9, go back and follow these steps again.)

 

Hope this helps!

 

Matt from way downstate New York