There are 15 line segments connecting the 6 points on the circle.
Method 1: Counting combinations
1. Choose any two points from the six points.
2. Draw a line segment connecting those two points.
3. Repeat steps 1 and 2 for all possible combinations of two points.
However, this method overcounts each line segment twice because we can choose the endpoints in either order. To correct for this, we need to divide the total number of combinations by 2.
Therefore, the number of line segments is:
Number of lines = 6C2 = 6! / (2! * 4!) = (6 * 5) / 2 = 15`
Method 2: Visualizing the problem
Imagine drawing all possible line segments connecting the six points. Each point will have three line segments emanating from it, connecting it to its three neighboring points on the circle. Therefore, the total number of line segments is:
`Number of lines = 6 points * 3 line segments/point = 18 line segments`
However, this method again overcounts each line segment twice. Therefore, we need to divide the total number of lines by 2:
`Number of lines = 18 line segments / 2 = 9 line segments`
This answer is incorrect because it only considers line segments connecting neighboring points. We also need to consider line segments connecting non-neighboring points on the circle.
Therefore, the correct answer using this method is:
`Number of lines = 6 points * (6 points - 1) / 2 = 6 * 5 / 2 = 15`
Both methods arrive at the same answer: there are 5 line segments that can be drawn connecting the six points on the circle.
Question:
The six points A, B, C, D, E, and F are located on a circle, as shown below. How many line segments having two of these points as endpoints can be drawn?