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Finding the Probability of forming a right triangle
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  1. Finding the Probability of forming a right triangle 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 ways three points can be chosen. MAKE a List! Remove all duplicates

  4. TOTAL: 20 Triangles. How many are Right Triangles? What is necessary to be guaranteed a right triangle? There are three diameters Line Segments AD, BE and CF

  5. TOTAL: 20 Triangles. How many are Right Triangles? How many right triangles can be formed with diameter AD?

  6. 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? How many right triangles can be formed with diameter CF?

  8. TOTAL: 20 Triangles. There are 12 right triangles Which triangles use diameters AD, BE, or CF?

  9. TOTAL: 20 Triangles. There are twelve right triangles Obtuse Obtuse Obtuse Equilateral Equilateral Obtuse Which triangles are equilateral triangles? Obtuse Obtuse Which triangles are 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) 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. Compare your results. Gather the results from the class. What does it show? Experimental Results

  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) =1 =3 =2 =2 =3 =1 Experimental Results

  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. Compare your results. Gather the results from the class. What does it show? =1 =3 =2 =2 =3 =1 Experimental Results

  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 A C B

  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? A There are 4 possible triangles BUT There are TWO Diameters, thus 4 Right Triangles D B C 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. 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? We will need to determine the total number of triangles that can be formed by using three points. How can we determine the total number of triangles?

  20. Method 1 for finding the total number of triangles: 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 of these triangles are right triangle? 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. How many right triangles can be formed with each diameter? A 6 right triangles B H How many right triangles can be formed? C G 24 P(right triangle)= ? D F Probability (right triangle) is 24/56 = 3/7

  24. How can we generalize how many right triangles will be formed?

  25. If n= number of points is an even number, can we determine the number of right triangles formed?

  26. Complete the chart for 10 and 12 points equally spaced around the circle.

  27. What patterns do you observe? What conjecture would you like to make?

  28. 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

  29. If we place nevenly 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?

  30. If we place nevenly 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: