Number of segments
Our goal with this lesson is to derive a formula for the number of segments between n points
To derive the formula, our strategy will be to see how many segments can be formed with 2, 3, 4, or 5 points
Then, we will try to identify a pattern that can help us derive the formula
How many segments can be formed with 2 points?
This is an easy question. We can get one segment
How many segments can be formed with 3 points?
I recommend not putting the 3 points on the same line. It will be easier to keep track and count the segments
Put the points on a piece of paper as shown below:
It is still easy to see that we can make 3 segments with 3 points
How many segments can be formed with 4 points? Again, do not put the points on the same line
Caution:
Usually people have no difficulties showing the 4 segments in blue. However, many people forget the two segments in red
So, 6 segments can be drawn with 4 points
How many segments can be formed with 5 points?
Things start getting a little complicated here. I will show you a way to count so you don't miss or overlook any segment
Keep in mind that the way I arrange the points is the way I believe will make it easier to count especially when you start counting the number of segments you can get with 5, 6, or 7 points
First, you can make these quick 5 segments that are shown in blue
Then, from each vertex, draw all the possibles diagonals
I am using a different color and a numbering system so you can clearly see
You can get two more from vertex #1 shown in red, 2 more from vertex #2 shown in green, and 1 more from vertex #3 shown in black
So 10 segments can be drawn with 5 points
Let us organize our findings. The table below will show you what we got so far and a math behind it
2 points


3 points


4 points


5 points


n points

n × ( n 1 )
/
2
segments

Explanations:
The denominator is always 2, so 2 will be the denominator in the general formula
The numerator has two numbers. The number on the right side of the multiplication is always 1 less the one on the left
That is why if the number on the left is n, the one on the right is n  1
What does n represent? Look carefully and you will see that it represents the number of points
Now that you have a formula, you can even calculate the number of segments you can get with 25 points if you like
25 × 24
/
2
= 300 segments
It is very useful to get a formula to get the number of segments with lots of points such as 25
If you tried to do the drawing above, things will get very messy
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