1. TRIANGLES, PROBABILITY, AND AMAZEMENT A CONNECTED EXPERIENCE FOR THE CLASSROOM JIM RAHN WWW.JAMESRAHN.COM JAMES.RAHN@VERIZON.NET

2. Question: If we place six(6) evenly spaced points around the circumference of a circle and then randomly select three points to form the vertices of a triangle, what is the probability that the triangle formed is a RIGHT TRIANGLE? Smith, Richard J., “Equal Arcs, Triangles, and Probability, Mathematics Teacher, Vol. 96, No. 9, December 2003, pp. 618-621.

3. Make a List of ALL Possible Triangles that can be formed using three of these points. MAKE a List! Triangle Vertices ABC ABD ABE ABF BCD BCE BCF BCA CDE CDF CDA CDB Triangle Vertices DEF DEA DEB DEC EFA EFB EFC EFD FAB FAC FAD FAE Remove all duplicates

4. TOTAL: 20 Triangles. How many are Right Triangles? There are three diameters Line Segments AD, BE and CF What is necessary to be guaranteed a right triangle? Triangle Vertices ABC ABD ABE ABF ACD ACE ACF ADE ADF AEF Triangle Vertices BCD BCE BCF BDE BDF CDE CDF CEF DEF EFB

5. Triangle Vertices ABC ABD ABE ABF ACD ACE ACF ADE ADF AEF TOTAL: 20 Triangles. How many are Right Triangles? How many right triangles can be formed with diameter AD? Triangle Vertices BCD BCE BCF BDE BDF CDE CDF CEF DEF EFB

6. Triangle Vertices BCD BCE BCF BDE BDF CDE CDF CEF DEF EFB Triangle Vertices ABC ABD ABE ABF ACD ACE ACF ADE ADF AEF TOTAL: 20 Triangles. How many are Right Triangles? How many right triangles can be formed with diameter BE?

7. TOTAL: 20 Triangles. How many are Right Triangles? Triangle Vertices ABC ABD ABE ABF ACD ACE ACF ADE ADF AEF Triangle Vertices BCD BCE BCF BDE BDF CDE CDF CEF DEF EFB How many right triangles can be formed with diameter CF?

8. TOTAL: 20 Triangles. Triangle Vertices Diameter End Points? ABC ABDAD ABEBE ABF ACDAD ACE ACFCF ADEAD ADFAD AEF Triangle Vertices Diameter End Points? BCD BCEBE BCFCF BDEBE BDF CDE CDFCF CEFCF DEF EFBBE Which triangles use diameters AD, BE, or CF? There are 12 right triangles

9. Triangle Vertices Diameter End Points? BCD BCEBE BCFCF BDEBE BDF CDE CDFCF CEFCF DEF EFBBE Triangle Vertices Diameter End Points? ABC ABDAD ABEBE ABF ACDAD ACE ACFCF ADEAD ADFAD AEF TOTAL: 20 Triangles. Which triangles are equilateral triangles? There are twelve right triangles Equilateral Which triangles are obtuse? Obtuse

10. If we place six(6) evenly spaced points around the circumference of a circle and then randomly select three points to form the vertices of a triangle, what is the probability that the triangle formed is a RIGHT TRIANGLE? Smith, Richard J., “Equal Arcs, Triangles, and Probability, Mathematics Teacher, Vol. 96, No. 9, December 2003, pp. 618-621.

11. Using simulation to determine the probability that the vertices of a right triangle is form by randomly selecting three points from six(6) evenly spaced points around the circumference of a circle. Place SIX Cubes (two of three different colors) into a bag. Draw out three cubes. If two cubes are of the same color, the triangle is a right triangle! ( Repeat 100 times) Right Triangle Non Right Triangle Total Experimental Results

12. Using simulation to determine the probability that the vertices of a right triangle is form by randomly selecting three points from six(6) evenly spaced points around the circumference of a circle. Right Triangle Non Right Triangle Total 5842100 Experimental Results Compare your results. Gather the results from the class. What does it show?

13. Using simulation to determine the probability that the vertices of a right triangle is form by randomly generating three numbers from three numbers. Opposite vertices will have the same numbers. Using your graphing calculator: Type randint(1, 3, 3). This means you will be selecting three numbers from 1,2, and 3. If two digits are the same number, the triangle is a right triangle! ( Repeat 100 times) Right Triangle Non Right Triangle Total Experimental Results =1 =2 =3

14. Using simulation to determine the probability that the vertices of a right triangle is form by randomly generating three numbers from three numbers. Opposite vertices will have the same numbers. Right Triangle Non Right Triangle Total 5842100 Experimental Results =1 =2 =3 Compare your results. Gather the results from the class. What does it show?

15. If we place three(3) evenly spaced points around the circumference of a circle and then randomly select three points to form the vertices of a triangle, what is the probability that the triangle formed is a RIGHT TRIANGLE? There is only 1 possible triangle and NO Diameters, Probability three points form a right triangle is 0 C B A

16. If we place four(4) evenly spaced points around the circumference of a circle and then randomly select three points to form the vertices of a triangle, what is the probability that the triangle formed is a RIGHT TRIANGLE? There are 4 possible triangles BUT There are TWO Diameters, thus 4 Right Triangles D C B A Probability three points form a right triangle is 4/4 = 1

17. If we place five(5) evenly spaced points around the circumference of a circle and then randomly select three points to form the vertices of a triangle, what is the probability that the triangle formed is a RIGHT TRIANGLE? There are 10 possible triangles BUT There are NO Diameters, thus NO Right Triangles Probability three points form a right triangle is 0

18. Number of equally spaced points (N) Total Number of Triangles (T) Number of Right Triangles (R) Probability a Triangle is Right (R/T) 3100/1 = 0 4444/4 = 1 51000/10 = 0 6201212/20 = 3/5 Odd > 3 0 8 10 12 : What patterns do you see in the Total Number of Triangles (T)?

19. If we place eight(8) evenly spaced points around the circumference of a circle and then randomly select three points to form the vertices of a triangle, what is the probability that the triangle formed is a RIGHT TRIANGLE? Number of Total Triangles How can we determine the total number of triangles? We will need to determine the total number of triangles that can be formed by using three points.

20. Method 1 for finding the total number of triangles: Number of equally spaced points (N) Total Number of Triangles (T) 311 441+3 5101+3+6 6201+3+6+10 81+3+6+10+? The total number of triangles is the sum of the first n-2 triangular numbers. The nth triangular number is

21. Method 2: Use combinations because we are choosing 3 points at random from 8 points. As long as we have the same three points selected there is only one triangle that can be formed.

22. How many right triangle can be found? What is necessary for the triangle to be a right triangle? One side of the triangle must be a diameter. How many diameters can be drawn? 4 Diameters

23. Probability three points form a right triangle is 24/56 = 3/7 Each diagonal forms 6 right triangle with the remaining vertices How many right triangles can be formed with each diameter? B A C D F G H How many right triangles can be formed? What is the probability of forming a right triangle when three points are selected at random from 8 points equally spaced around a circle?

24. Number (N) of equally spaced points Total Number (T) of Triangles Number (R) of Right Triangles Probability a Triangle is Right (R/T) 3100/1 = 0 4444/4 = 1 51000/10 = 0 6201212/30 = 3/5 Odd > 3 0 8562424/56 = 3/7 10 12 : How can we generalize how many right triangles will be formed?

25. What must be true about the number of points equally spaced around the circle? If n= number of points is an even number, can we determine the number of right triangles formed?

26. Number (N) of equally spaced points Total Number (T) of Triangles Number (R) of Right Triangles Probability a Triangle is Right (R/T) 3100/1 = 0 4444/4 = 1 51000/10 = 0 6201212/30 = 3/5 Odd > 3 0 8562424/56 = 3/7 10 12 : Complete the chart for 10 and 12 points equally spaced around the circle.

27. 10 points equally spaced around a circle Number of right triangles Number of total triangles Probability of forming a right triangle

28. 12 points equally spaced around a circle Number of right triangles Number of total triangles Probability of forming a right triangle

29. Number (N) of equally spaced points Total Number (T) of Triangles Number (R) of Right Triangles Probability a Triangle is Right (R/T) 3100/1 = 0 4444/4 = 1 51000/10 = 0 6201212/30 = 3/5 Odd > 3 0 8562424/56 = 3/7 104012040/120=3/9 128022060/220=3/11 : What patterns do you observe? What conjecture would you like to make?

30. If n points are equally spaced on the circumference of a circle and if three points are chosen at random, the probability that the three points will form a right triangle is

31. If we place n evenly spaced points around the circumference of a circle, where n is an even number greater than 3, and then randomly select three points to form the vertices of a triangle, what is the probability that the triangle formed is a RIGHT TRIANGLE? What do you notice about the number of possible triangles? Number of points around the circle Number of Triangles Using Triangular Numbers Using Combinations 441+3 4 C 3 =4 6201+3+6+10 6 C 3 =20 8561+3+6+10+15+21 8 C 3 =56 101201+3+6+10+15+21+28+36 10 C 3 =120 122201+3+6+10+15+21+28+36+ 45+55 12 C 3 =220

32. Number of points around the circle Number of Triangles Using Triangular Numbers Using Combinations 441+3 4 C 3 =4 6201+3+6+10 6 C 3 =20 8561+3+6+10+15+21 8 C 3 =56 101201+3+6+10+15+21+28+36 10 C 3 =120 122201+3+6+10+15+21+28+36+ 45+55 12 C 3 =220 n n-2 terms

33. If we place n evenly spaced points around the circumference of a circle, where n is an even number greater than 3, and then randomly select three points to form the vertices of a triangle, what is the probability that the triangle formed is a RIGHT TRIANGLE? How many right triangles will there be? There are n/2 diameters. Each forms with ( n - 2 ) right triangles with the remaining vertices. Thus the number of right triangles is:

34. Number (N) of equally spaced points Total Number (T) of Triangles Number (R) of Right Triangles Probability a Triangle is Right (R/T) 3100/1 = 0 4444/4 = 1 51000/10 = 0 6201212/30 = 3/5 Odd > 3 0 8562424/56 = 3/7 101204040/120 = 3/9 122206060/220 = 3/11 : Even (E) number>3 3/(E – 1)