Chapter 10 Section 10.1 Pythagorean Theorem

Pythagorus Pythagoras lived about 600 years before Christ (BC) He was one of the first Greek mathematical thinkers. He lived in the Greek colonies of Sicily. He had a group of “disciples” who followed him around and taught others what he taught them. The “Pythagoreans” were known for their pure lives (they didn't eat beans, for example, because they thought beans weren’t pure enough). They wore their hair long, and wore only simple clothing, and went barefoot. Pythagoreans were interested in philosophy, but especially in music and mathematics; two ways of making order out of chaos. Music is noise that makes sense, and mathematics is rules for how the world works. Pythagoras himself is best known for proving that the Pythagorean Theorem was true. The Sumerians, two thousand years earlier, already knew it was true and used it in their measurements, but Pythagoras was credited as proving it to be true. You will study the Pythagorean Theorem IN DEPTH in Geometry next year!

The Pythagorean Theorem is one of the most famous theorems in mathematics. The relationship it describes has been known for thousands of years a2 + b2 = c2

Pythagorean Theorem Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. b a c a 2 + b 2 = c 2 11/21/2018 Clark 10.1

Common Pythagorean Triples Patterns Pythagorean Triple A set of three positive integers a, b, and c that satisfy the equation a 2 + b 2 = c 2. For example, the integers 3, 4, and 5 form a Pythagorean triple because 32 + 4 2 = 5 2 . Common Pythagorean Triples ( 3, 4, 5) ( 5, 12, 13) ( 7, 24, 25) ( 8, 15, 17) ( 9, 40, 41) (11, 60, 61) (12, 35, 37) (13, 84, 85) (16, 63, 65) (20, 21, 29) (28, 45, 53) (33, 56, 65) (36, 77, 85) (39, 80, 89) (48, 55, 73) (65, 72, 97) 11/21/2018 Clark 10.1

Finding the length of a hypotenuse 12 x 5 Find the length of the hypotenuse of the right triangle. (hypotenuse)2= (leg)2 + (leg)2 Pythagorean Theorem x 2 = 5 2 + 12 2 x 2 = 25 + 144 x 2 = 169 x = 13 Because the side lengths 5, 12, and 13 are integers, they form a Pythagorean triple.

What if you have a radical? Using the Pythagorean Theorem 4 x 20 What if you have a radical? (hypotenuse)2 = (leg)2 + (leg)2 Pythagorean Theorem x 2 = 20 2 + 4 2 x 2 = 20 + 16 x 2 = 36 x = 6 Why would you not need to answer with 6 and -6? Lengths can’t be negative!

Finding the length of a leg Find the length of the leg of the right triangle (hypotenuse)2 = (leg)2 + (leg)2 Pythagorean theorem 10 2 = x 2 + 8 2 x 10 8 100 = x 2 + 64 36 = x 2 36 = x 6 = x 11/21/2018 Clark 10.1

Finding the missing length (hypotenuse)2 = (leg)2 + (leg)2 13 2 = 12 2 + x 2 169 = 144 + x 2 25 = x 2 25 = x 5= x 11/21/2018 Clark 10.1

Finding the missing length (hypotenuse)2 = (leg)2 + (leg)2 13 2 = 12 2 + x 2 169 = 144 + x 2 25 = x 2 25 = x 5= x 11/21/2018 Clark 10.1

SUPPORT BEAM These skyscrapers are connected by a skywalk with support beams. You can use the Pythagorean Theorem to find the approximate length of each support beam. 11/21/2018 Clark 10.1

The length of each support beam is about 52.95 meters. Each support beam forms the hypotenuse of a right triangle. The right triangles are congruent, so the support beams are the same length. 23.26 m 47.57 m x support beams x 2 = (23.26)2 + (47.57)2 x = (23.26)2 + (47.57)2 x  52.95 The length of each support beam is about 52.95 meters. 11/21/2018 Clark 10.1

ABC is a right triangle c 2 = a 2 + b 2 THEOREM In a triangle, if c2 = a2 + b2, then the triangle is a right triangle A B C b a c ABC is a right triangle c 2 = a 2 + b 2 11/21/2018 Clark 10.1

Determine whether the given lengths can be side lengths of a right triangle. A. 12 cm, 16 cm, 20 cm B. 8 m, 10 m, 13 m   11/21/2018 Clark 10.1

When would we use this? Using the Pythagorean Theorem John leaves school to go home. He walks 6 blocks North and then 8 blocks west. How far is John from the school? The distance from school to home is the length of the hypotenuse. Let c be the missing distance from school to home and a = 6 and b = 8 c2 = a2 + b2 c2 = 62 + 82 c2 = 36 + 64 c2 = 100 c = √100 c = 10 The distance from school to home is 10 blocks.

When would we use this? a2 + b2 = c2 242 + 362 = c2 576 + 1296 = c2 Using the Pythagorean Theorem When would we use this? TV’s are measured diagonally. If your home TV is 24 inches tall and 36 inches wide, what is the length diagonally of your TV? 36” a2 + b2 = c2 24” 242 + 362 = c2 576 + 1296 = c2 1872 = c2 43.266 = c The TV is about 43.3 inches