Number of Diagonals of an n-sided Polygon

Maximum Number of Regions Formed by the Circle and n Chords

Given a circle and points on the circle. With n points on the circle

1. What is the number of chords determined by n points?

2. What is the number of diagonals determined by n points? See

Triangular Numbers Problems, Problem 4.3. What is the maximum number of regions determined by the chords and the circle rim.

See the image to the right, depiccting cases for 1 through 7 points.

NOTE: Question 1 above is equivalent to the

Handshake Problem(Problem 3): Each chord corresponds to a handshake between two people.ALSO: Since each chord corresponds to a pair of points, Question 1 is equivalent to finding the

number of combinations of n things taken 2 at at time.For Question 3, above, it might be useful to use

colorsin a systematic way to count the regions as the number of points increases.Click HERE for a

JavaGSPfile to show building a table of values for each of the counts.If JavaGSP does not work, click

HEREfor a GSP file and follow the directions on the page.

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