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Section 9.6 Families of Right Triangles

Section 9.6 Families of Right Triangles. By: Maggie Fruehan. Any three whole numbers that satisfy the equation a ²+b²=c² form a Pythagorean Triple. 15. 6. 5. 8. 15. 3. 12. 20. 10. 25. 9. 4. These four triangles are all members of the (3,4,5) family. Some History….

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Section 9.6 Families of Right Triangles

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

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

  3. Some History… • The study of these Pythagorean triplesbegan 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. The number of spaces match the (3,4,5) triple! Sting pulled taut String with 12 knots

  4. Pythagorean Triples must appear as whole numbers. 0.5 2½ 0.4 1½ 0.3 2 Even though these are not families, they all are members of the (3,4,5) family.

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

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

  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. • Solve for the missing side of the easier triangle. • Convert back to the original problem. The family is (8,15,17). Thus, 2x=17 and x=8½ (in the original problem). x 2x 4 2(4)=8 2(7½)=15 7½

  8. More Reduced Triangles! 2² + y² = 3² 4 + y² = 9 y² = 5 y = 600 400 3 *Make sure to change the variable! 2 y x = You can also enlarge the triangle! y² = 5² + 8² y² = 89 y = y x 1¼ 5 2 8

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

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

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

  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/frintch2ch3.pdf • http://itsatrap.net/

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