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

Apr 02, 20 12:05 PM
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