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Pythagorean Theorem Obj: SWBAT identify and apply the Pythagorean Thm and its converse to find missing sides and prove triangles are right Standard: M11.C.1.4.1 Find the measure of a side of a right triangle using the Pythagorean Thm History of the Pythagorean Thm At the height of their power, nearly a millennium before Pythagoras, circa 1900 1600 BCE , the Babylonians (Babylon located in modern day Iraq) identify what are now called Pythagorean triples (a set of positive integers a, b, c such that a2 + b2 = c2 History of the Pythagorean Thm A Chinese astronomical and mathematical treatise called the Chou Pei Suan Ching (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven, ca. 500-200 B.C.), possibly predating Pythagoras, gives a statement of and geometrical demonstration of the Pythagorean Thm. History of the Pythagorean Thm Despite evidence predating him, the Greek named Pythagoras is credited with the theorem. According to tradition, Pythagoras once said, “Number rules the universe…” WHAT A FREAKING GENIUS!!!! Basics of the Right Triangle hypotenuse leg leg Right angle Pythagorean Thm In ANY right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a2 + b 2 = c 2 Use the Pythagorean Thm to solve for x. x 3 4 1. Square both legs 1 2 3 4 5 6 7 8 9 10 11 12 4 ft 13 14 15 4ft 16 3 ft 1 2 3 4 5 6 7 8 9 3 ft 2. Count the total squares 1 2 3 4 5 6 7 8 9 10 11 12 4 ft 13 14 15 4ft 16 9 + 16 = 25 3 ft 1 2 3 4 5 6 7 8 9 3 ft 3. Put that number of squares on the hypotenuse 21 16 1 5 9 13 22 11 17 23 24 6 12 18 7 13 19 25 1 2 3 4 8 14 20 2 3 15 6 7 8 9 4 10 10 11 12 4 5 ft 14 15 16 3 4ft 1 ft2 3 9 + 16 = 25 4 5 6 7 8 9 3 ft 4. Count the number of squares that touch the hypotenuse. 21 16 1 5 9 13 22 11 17 23 24 6 12 18 7 13 19 25 1 2 3 4 8 14 20 2 3 15 6 7 8 9 4 10 10 11 12 4 5 ft 14 15 16 3 4ft 1 ft2 3 9 + 16 = 25 #=5 4 5 6 7 8 9 3 ft 5.That number is the length of the hypotenuse. 21 16 1 5 9 13 22 11 17 23 24 6 12 18 7 13 19 25 1 2 3 4 8 14 20 2 3 15 6 7 8 9 4 10 10 11 12 4 5 ft 14 15 16 3 #=5 4ft 1 ft2 3 Length = 5 9 + 16 = 25 4 5 6 7 8 9 3 ft LOOK FAMILIAR?! WOW, it’s actually the Pythagorean Thm! That’s so freaking cool!!! Why are the wrong answers wrong??? Pythagorean Triples Are short cuts! They are sets of 3 whole numbers (a, b, and c) that satisfy the equation a2 + b2 = c2 Most frequent examples: *** 3, 4, 5 (where a=3, b=4, c=5) *** 5, 12, 13 8, 15, 17 7, 24, 25 ANY scale or multiple of Pythagorean triples will work!!! Pythagorean Triple Example Don’t be fooled by the disguise… ...it’s still the Pythagorean Thm!! Pythagorean Thm application with even more Geometry!!! It’s actually that much more fun!!! HOLY SMOKES, LOOK AT ALL OF THIS GEOMETRY!!! A-MAZ-ING!!!!