1. Section 9.6 Section 9.6 Families of Right Triangles By: Maggie Fruehan

2. Pythagorean Triple Any three whole numbers that satisfy the equation a²+b²=c² form a Pythagorean Triple. 3 4 5 6 8 10 15 12 9 15 20 (3,4,5) These four triangles are all members of the (3,4,5) family. 25

3. { The study of these Pythagorean triples began long before the time of Pythagoras. { There are Babylonian tablets that contain lists of such triples. { Pythagorean triples were also used in ancient Egypt. For example, to produce a right angle they took a piece of string, marked it into 12 equal segments, tied it into a loop, and held it taut in the form of a (3,4,5) triangle. Sting pulled taut String with 12 knots The number of spaces match the (3,4,5) triple! Some History…

4. must Pythagorean Triples must appear as whole numbers. 1½1½ 2 2½2½ Even though these are not families, they all are members of the (3,4,5) family. 0.3 0.5 0.4

5. infinitely There are infinitely many families, but the most frequently seen are the: (3,4,5) (5,12,13) (7,24,25) 27 ? 45 45/9= 5 27/9= 3 ? /9= 4 ? =36 72 30 ? 72/6= 12 30/6= 5 ? /6= 13 ? =78 2.5 0.7 ? 2.5(10)= 25 0.7(10)= 7 ? (10)= 24 ? = 2.4

6. More Families… (8,15,17) (9,40,41) 1½1½ ? 2 1½ = 2 = ? = 80 ? 18 18/2= 9 80/2= 40 ?/ 2= 41 ? = 82 Numerators resemble the (3,4,5) triangle!

7. The Principal of the Reduced Triangle Reduce the difficulty of the problem by multiplying or dividing the three lengths by the same number to obtain a similar, but simpler, triangle in the same family.  Reduce the difficulty of the problem by multiplying or dividing the three lengths by the same number to obtain a similar, but simpler, triangle in the same family. { Solve for the missing side of the easier triangle. { Convert back to the original problem. 4 7½7½ x 2(4)=8 2(7½)=15 2x The family is (8,15,17). Thus, 2x=17 and x=8½ (in the original problem).

8. More Reduced Triangles! 400 600 x = 2 3 y *Make sure to change the variable! 2² + y² = 3² 4 + y² = 9 y² = 5 y = You can also enlarge the triangle! 1¼1¼ 2 x y² = 5² + 8² y² = 89 y = y 5 8

9. 15 17 6 26 25 x Find x. x = 7

10. x 15 13 25 x =12 Find x Square x 40 41 Find x x =

11. http://www.itsatrap.net/ 8 6 x HA!!! IT’S NOT 10!!! because it’s a… 6² + x² = 8² 36 + x² = 64 x² = 28 x =

12. Works Cited  Richard Rhoad, George Milauskas, Robert Whipple, Geometry for Enjoyment and Challenge. Evantson, Illinois: McDougal, Littell & Company, 1991.  http://www.math.brown.edu/~jhs/frintc h2ch3.pdf http://www.math.brown.edu/~jhs/frintc h2ch3.pdf  http://itsatrap.net/ http://itsatrap.net/