Pythagorean Theorem By: Carrie Yates 8 th Grade Mathematics

Who is Pythagoras? Who is Pythagoras? Born between 580 and 572 BC and died between 500 and 490 BCBorn between 580 and 572 BC and died between 500 and 490 BC Ionian Greek mathematicianIonian Greek mathematician Founder of the religious movement PythagoreanismFounder of the religious movement Pythagoreanism Revered as a great mathematician, mystic and scientistRevered as a great mathematician, mystic and scientist Known as “the father of numbers”Known as “the father of numbers”

History Born on Samos, a Greek islandBorn on Samos, a Greek island Undertook a reform of the cultural life of Croton, urging the city to follow virtue and form an elite circle of followers called PythagoreansUndertook a reform of the cultural life of Croton, urging the city to follow virtue and form an elite circle of followers called Pythagoreans Opened a school for male and females - they were called MathematikoiOpened a school for male and females - they were called Mathematikoi They lived at the school with no personal possession and were required to assume a mainly vegetarian diet.They lived at the school with no personal possession and were required to assume a mainly vegetarian diet. Other students in neighboring areas were also permitted to attend but were known as Akousmatikoi and they were permitted to have personal belongings and eat meat.Other students in neighboring areas were also permitted to attend but were known as Akousmatikoi and they were permitted to have personal belongings and eat meat.

Historycontinued Pythagoras’s followers were commonly called “Pythagoreans”Pythagoras’s followers were commonly called “Pythagoreans” Pythagoreans followed a structured life of religious teaching, common meals, exercise, reading and philosophical studyPythagoreans followed a structured life of religious teaching, common meals, exercise, reading and philosophical study Members were required to live ethically, love one another, share political beliefs, practice pacifism, and devote themselves to the mathematics of natureMembers were required to live ethically, love one another, share political beliefs, practice pacifism, and devote themselves to the mathematics of nature The organization was in some ways a school, a brotherhood and a monasteryThe organization was in some ways a school, a brotherhood and a monastery

History continued Pythagoreans elaborated on a theory of numbersPythagoreans elaborated on a theory of numbers One of Pythagoras’s beliefs was that the essence of being is numberOne of Pythagoras’s beliefs was that the essence of being is number

Contributions to Mathematics Believed all things can be reduced to number relations He came up with: Different ways to classify them Even and odd numbers, perfect numbers, triangular numbers, square numbers and properties of numbers Theory of irrational numbers and construction of cosmic figure

Contributions related to Geometry ► Figured out sum of triangle equals the sum of two right triangles ► Solved the constructing figures of a given area and geometrical algebra and first three regular solids and maybe all five ► Discovered square numbers

LARGEST CONTRIBUTION Theorem  Hypotenuse is equal to the sum of the squares of the other two sides  (a(a)=b(b)=c(c))

Other Accomplishments Discovery that music was based on proportional intervals of the numbers one through four Discovery that music was based on proportional intervals of the numbers one through four Discovered that musical notes could be translated into mathematical equations Discovered that musical notes could be translated into mathematical equations

Pythagoream Thereom Algebraic expression c is the hypotenuse a and b are the legs of the triangle Hypotenuse – longest side of a right triangle; side opposite right triangle

Important Information Pythagorean Theorem Pythagorean Theorem Can only be used on a right triangles Can only be used on a right triangles The legs are ALWAYS the sides adjacent to the 90 degree angle The legs are ALWAYS the sides adjacent to the 90 degree angle Does not matter which leg is labeled a or labeled b Does not matter which leg is labeled a or labeled b c is ALWAYS the hypotenuse c is ALWAYS the hypotenuse

Quick Review  Which number represents the hypotenuse (c)?  Which two numbers represents the legs (a and b)?

Answers:  Question 1 : hypotenuse = 12  Question 2 : legs = 7 and 9

Plug in the Values:  a² + b² = c² If …  a = 9  b = 7  c = 12 Then … 9² + 7² = 12²

Solve for Hypotenuse  a² + b² = c² If…  a = 4  b = 3  c = x Then…  4² + 3² = x²

Answer for Hypotenuse  4² + 3² = x²  = x²  25 = x²  √25 = √x  5 = x

Solve for the missing leg: a² + b² = c² If … a = 7 b = x c = 11 Then … 7² + x² = 11²

Answer for the leg (x): 7² + x² = 11² 49 + x² = 121 Subtract 49 from both sides x² = 72 √x = √72 x = 8.49

LET US HAVE SOME FUN Practice Problems PracPyth.htm PracPyth.htm theorem-game.php theorem-game.php

Class discussion The End