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

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

{ 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…

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

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 ? = ? 72/6= 12 30/6= 5 ? /6= 13 ? = ? 2.5(10)= (10)= 7 ? (10)= 24 ? = 2.4

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!

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).

More Reduced Triangles! 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

x Find x. x = 7

x x =12 Find x Square x Find x x =

8 6 x HA!!! IT’S NOT 10!!! because it’s a… 6² + x² = 8² 36 + x² = 64 x² = 28 x =

Works Cited  Richard Rhoad, George Milauskas, Robert Whipple, Geometry for Enjoyment and Challenge. Evantson, Illinois: McDougal, Littell & Company,  h2ch3.pdf h2ch3.pdf 