Regular Polygons and Schedules 831
BIG IDEA An easy, yet not obvious way to schedule teams in a tournament involves chords in a circle.
Name all the types of fi gures that can be intersections of a right cylinder and a plane.
In this lesson, we show an application of regular polygons and circles to a discrete geometry situation in which points stand for teams.
A Round-Robin Tournament
In many tournaments or leagues, from basketball to chess to bowling, each competitor or team plays all the others. Each time as many teams as possible are playing, the set of matches is called a round. When each competitor plays each other competitor exactly once, it is called a round-robin tournament. Scheduling a round-robin tournament can be tricky. Suppose there are 7 teams to be scheduled so that each plays the other six. The ﬁ rst thing you might do is number the teams 1 through 7. Because there is an odd number of teams, in each round, one team doesn’t play. That team gets a bye.
Round-robin schedules can be created in many ways. One example of an incomplete round-robin schedule for seven teams is given. The table lists who each team is playing in each round. For example, since team 1 is paired with team 2 in Round 1, a “2” is placed in the team-1 cell and a “1” is placed in the team-2 cell. Copy the table into your notebook. Complete the table allowing each team to play each other team exactly once.
Team 1 Team 2 Team 3 Team 4 Team 5 Team 6 Team 7
Round 1 2 1 ? ? ? ? bye Round 2 3 ? 1 ? ? bye ? Round 3 4 ? ? 1 bye ? ? Round 4 5 ? ? bye 1 ? ? Round 5 6 ? bye ? ? 1 ? Round 6 7 bye ? ? ? ? 1 Round 7 bye ? ? ? ? ? ? SMP_SEGEO_C14L02_831-836.indd 831 SMP_SEGEO_C14L02_831-836.indd 831 5/27/08 10:27:48 AM5/27/08 10:27:48 AM
832 Further Work with Circles
Using Regular Polygons to Schedule
an Odd Number of Teams
In the Activity, you are asked to schedule a round-robin tournament for seven teams. What if there were more teams? It would be nice if there was some algorithm that automatically created the schedule. The algorithm described here is surprising in that it uses properties of regular polygons and circles. The example below is given for 9 teams.
Step 1 Let the 9 teams be represented by vertices of a regular 9-gon (nonagon). By the Center of a Regular Polygon Theorem, we know we can place the vertices equally spaced on a circle for convenience. Copy the fi gure at the right.
Step 2 (the fi rst round) Draw a chord and all chords parallel to it connecting pairs of numbered points. Because the polygon has an odd number of sides, the minor arcs of each chord pictured have different measures. So no two chords have the same length. The endpoints of these chords are the fi rst round’s schedule.
First round: 2-9 3-8 4-7 5-6 1-bye
Because the top chord connects the numbers 2 and 9, team 2 plays team 9 in the fi rst round. This is called a pairing, and is written 2-9. Also in the fi rst round, team 3 plays team 8, team 4 plays team 7, team 5 plays team 6, and team 1 gets a bye. The full schedule will be completed when all sides and diagonals of the nonagon have been drawn.
Step 3 (the second round) Rotate the chords __19 of a revolution. For example, the chord pairing 2-9 for the fi rst round rotates into the pairing 1−8 for the second round.
Second round: 1-8 2-7 3-6 4-5 9-bye Step 4 Continue rotating __19 of a revolution for each round. Because in a
round no two chords have the same length, no pairing repeats. (For example, look at the chord that forms the pair 3-8 in round 1. No other chord has the same length, so no other chord will pair team 3 with team 8.)
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Regular Polygons and Schedules 833 In a total of nine rounds, the schedule is complete. We leave the 3rd through
9th rounds for you to fi gure out.
Third round: 9-7 1-6 2-5 3-4 ? Fourth round: ? ? ? ? ? Fifth round: ? ? ? ? ? Sixth round: ? ? ? ? ? Seventh round: ? ? ? ? ? Eighth round: ? ? ? ? ? Ninth round: ? ? ? ? ?
In a nonagon, there are 27 diagonals and 9 sides, making a total of 36 segments. Each diagonal can be placed in one of three groups based on its length.
With 9 teams there are 4 games for each of 9 rounds, or 36 pairings. Of these, 9 are sides of the regular nonagon, 9 are congruent shortest diagonals, 9 are congruent middle-length diagonals, and 9 are congruent longest diagonals. Thus, the number of pairings in a round robin with n teams is equal to n plus the number of diagonals. How would you determine the number of diagonals in an n-gon? If n is small, counting the diagonals would be an easy task. But what if n was a large number? Activity 3 leads you to an algebraic expression for the number of diagonals of an n-gon.
The diagram at the right shows that, from a given vertex of a hexagon, it is possible to draw 3 diagonals.
Step 1 Complete the table below by sketching each polygon and counting. We have done the hexagon for you.
number of sides 4 5 6 7 8 . . . n
number of vertices ? ? ? ? ? ?
number of diagonals
from each vertex ? ? ? ? ? . . . ?
Step 2 Why is the number of diagonals from each vertex always 3 fewer than the number of vertices?
Step 3 Create an expression for the total number of diagonals that can be drawn in the polygon from all of the vertices.
Step 4 Use your expression to calculate the total number of diagonals in a hexagon. Then count the diagonals and verify that your expression gave you the correct number.
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834 Further Work with Circles
An expression that you might have developed in the Activity is stated in the theorem below.
Diagonals in a Polygon Theorem
A polygon with n sides has _______n(n - 3)
Proof In a polygon with n sides, there are n vertices. From each vertex, (n - 3) diagonals can be drawn. The product n(n - 3) is divided by 2 because the product counts every diagonal twice.
A polygon has 20 diagonals. How many sides does it have?
Solution In an n-gon, there are n(n _______ -2 3) diagonals. So, n(n - 3) _______ 2 = 20 n(n - 3) = 40 n(n - 3) - 40 = 40 n2- 3n - 40 = 0 (n - 8)(n + 5) = 0 n - 8 = 0 or n + 5 = 0 n = 8 or n =–5.
Since a polygon cannot have a negative number of sides, the polygon with 20 diagonals has 8 sides. It is an octagon.
Scheduling an Even Number of Teams
The procedure for scheduling an odd number of teams will not work with an even number of teams. If parallel chords are drawn using the vertices of a regular decagon, some will have the same length. You can see that as you rotate, you will repeat pairings.
However, instead of putting the tenth team on the circle, it can be placed at the center of the circle. The radius joins team 10 and the team receiving the bye in the schedule for 9 teams. As you rotate the chords to make the schedule, rotate the radius too! This shows the surprising result: it takes as many rounds for a schedule of 9 teams as it does for a schedule
of 10 teams. In general, when n is even, it takes as many weeks for a schedule of n - 1 teams as for a schedule of n teams.
How many diagonals does a 27-gon have? 9 10 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 10 SMP_SEGEO_C14L02_831-836.indd 834 SMP_SEGEO_C14L02_831-836.indd 834 5/27/08 10:27:57 AM5/27/08 10:27:57 AM
Regular Polygons and Schedules 835 In the Questions, you should assume that all schedules are round-robin.
COVERING THE IDEAS 1. Refer to Activity 2.
a. What teams are paired to play in round 6? b. What team plays team 7 in the 5th round? c. In what round does team 4 play team 5?
2. Refer to the use of the regular nonagon to schedule 9 teams. a. How is each team represented?
b. How is a game between teams 3 and 6 represented?
c. What is the relationship between all the chords drawn for a given round?
d. How are the pairings of one week related to those of the next week?
e. What are the pairings in the 9th round of play?
3. A regular nonagon has diagonals of how many different lengths? 4. When an odd number of teams are scheduled, how many byes
are in each round?
5. How many rounds of play are needed to accommodate 32 teams? 6. How is the number of pairings in a round robin of n teams
related to the number of diagonals of an n-gon? 7. How many diagonals does a 15-gon have?
8. According to the Diagonals of a Polygon Theorem, how many diagonals does a triangle have? Is that correct?
9. If there is an even number of teams in the tournament, how does the algorithm for creating the schedule change?
10. True or False It takes as many weeks for a schedule of 23 teams as it does for a schedule of 24 teams.
11. Make a complete schedule for a tournament with 6 teams. 12. A polygon has 77 diagonals. How many sides does it have?
APPLYING THE MATHEMATICS
13. Explain why you cannot begin with the diagram at the right and rotate if you wish to schedule 8 teams.
14. Shreya is scheduling a 20-team round robin. a. How many rounds will it take?
b. Explain why there are no byes.
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836 Further Work with Circles
15. Multiple Choice Which two of the following descriptions of a point are used in the idea of scheduling?
A dot B ordered pair
C location D node of network
16. A round-robin tournament has 66 pairings. How many teams are in the tournament?
17. Each interior angle of a regular polygon measures 135º. Find the number of diagonals of the polygon.
18. In the ﬁ gure at the right, each point on the circle represents a person and each chord represents a handshake.
a. What is the minimum number of handshakes needed for each person in a group of 9 to shake hands with every other person? b. What is the minimum number of handshakes needed for each
person in a party of n to shake hands with every other person? REVIEW
19. Suppose a chord has length 99 in a circle with diameter 100. How far is the chord from the center of the circle? (Lesson 14-1) 20. Refer to the ﬁ gure at the right. (Lessons 12-7, 6-3)
a. Name two angles whose intercepted arc is AB .
b. What can you say about the measure of these angles?
c. Name two angles whose intercepted arc is CD . What can you
say about the measure of these angles?
d. Explain why CEA ˜ DEB.
21. The circle at the right has center O, radius r, and m∠O = 90. Write an expression for the area of the circle outside of OBC. (Lessons 8-9, 8-4)
22. How many of the diagonals of a regular 19-gon are on lines of symmetry? (Lesson 6-8)
23. State the Exterior Angle Theorem. (Lesson 5-7)
24. a. Find a schedule for teams in a league involving your school or community.
b. What factors affect schedules that this lesson does not mention?
25. Explain why, other than a quadrilateral, no polygon has a number of diagonals that is an integer power of 2.
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