Who was Pythagoras? An ancient Greek thinker who was fond of poetryand literature He himself was not a geometer His followers, dubbed the PythagoreanBrotherhood, were a religious cult Contrary to popular belief, Pythagoras was NOTthe founder of the Pythagorean Theorem The Pythagorean Brotherhood founded the theoremover 100 years after Pythagoras had died
What It Relates to: Any and all right triangles This means that this angle is a right angle A right angle = 90º Since there are 180º in a triangle, the other 2 angles must addup to 90º
What Is It? Let’s label the triangle by its angles ABC If the triangle’s angles are ABC, then the sidesopposite those angles are a, b, and c,respectivelyACBbac
What Is It? a and b are the sides, or legs, of the righttriangle c is the hypotenuse, or the side opposite theright angle, which is always the longest side ofthe right triangle In any triangle, the sums of any 2 sides isgreater than the length of the 3rdside, so: a + b > c a + c > b b + c > a
What Is It? However, sometimes when we square the sides,the sum of the squares is equal to the square ofthe hypotenuse a2+ b2= c2 This is the Pythagorean Theorem The Pythagorean Theorem states: In any right triangle, the sum of the squares of the 2 sides isequal to the square of the hypotenuse. Conversely, if the sum of the squares of the 2 sides is equalto the square of the hypotenuse, then you have a righttriangle.
What Is It? What does this mean?It means that, if you have a right triangle, youknow that the squares of the 2 sides willalways add up to the square of thehypotenuse.It means that if you have a triangle in whichthe squares of the 2 sides add up to thesquare of the hypotenuse, you know you havea right triangle.
What Is It? Let’s look at some proofs:1 2 3 45 6 7 89 10 11 1213 14 15 161 2 34 5 67 8 9
What Is It? If you add up all of the little squares, i.e. 9+ 16, you get 25. This is in accord with 32+ 42= 52 Here, we see that the lengths of the sidesare 3 and 4, and the length of thehypotenuse is 5. Using the Pythagorean Theorem, we get32+ 42= 52, or 9 + 16 = 25.
What Is It? What is the area of the large blue square? By multiplying the length by the width, we get (a + b)2 Simplify to a2+ 2ab + b2 By adding up the areas of 4 small blue triangles andthe 1 small purple square, we get, 4(1/2)ab + c2 Simplify to 2ab + c2 If we set the areas equal to each other, we get a2+2ab + b2= 2ab + c2 The 2ab’s cancel out, so we get a2+ b2= c2 Thus, the Pythagorean Theorem has been onceagain proven.
What Is It? The Pythagorean Theorem specifically refers tosquares of the sides, however any 2-dimensional relation between the 2 sides andthe hypotenuse would work If we drew circles on each side and the hypotenuse,with the sides as the diameter of each respectivecircle, then the areas of the circles on the 2 sides willsum up to the area of the circle on the hypotenuse Algebraically, this is seen as ka2+ kb2= kc2since kcan be factored out, where k is some constant
Practice Find the missing lengths: a = 3, b = 4, c = ? a = 7, b = ?, c = 25 a = ?, b = 12, c = 13 Are triangles with the following lengths righttriangles? a = 7, b = 8, c = 9 a = 12, b = 16, c = 20 a = 11, b = 58, c = 61
Practice To get from point A to point B you must avoidwalking through a pond. To avoid the pond, youmust walk 34 meters south and 41 meters east.To the nearest meter, how many meters wouldbe saved if it were possible to walk through thepond? A baseball diamond is a square with sides of 90feet. What is the shortest distance, tothenearest tenth of a foot, between first baseand third base?
Practice In a computer catalog, a computer monitor islisted as being 19 inches. This distance is thediagonal distance across the screen. If thescreen measures 10 inches in height, what is theactual width of the screen to the nearest inch? Oscars dog house is shaped like a tent. Theslanted sides are both 5 feet long and thebottom of the house is 6 feet across. What isthe height of his dog house, in feet, at its tallestpoint?
Why Does It Matter? The Pythagorean Theorem allows us to do manythings in real-life situations. The professional fields in which it is useful are: Civil Engineering Construction Astronomy Physics Particle Physics Advanced Mathematics Ancient Warfare
Why Does It Matter? Civil Engineering:Building bridgesMeasuring distances across rivers in order todetermine the lengths of proposed bridgesBuilding foundations of skyscrapers ConstructionMeasuring anglesEnsuring solid and level foundations
Why Does It Matter? AstronomyMeasuring distances in a 3-dimensionalspaceCalculating shadows cast by astronomicalbodies PhysicsDetermining pressure in bridge constructionUnderstanding ramps, levers, and screws
Why Does It Matter? Particle Physics Calculating distances of particles in 3-dimensional space Advanced Mathematics Pythagorean Triples Trigonometry and the Unit Circle Vectors Calculating distances between points on a Cartesian Plane Ancient Warfare The Ancient Romans used it to measure the distance thatcatapults had to be from their target
Food for Thought The distance formula for 2 points on a Cartesian Plane is derivedfrom the Pythagorean Theorem The distance formula is d = √[(x2 – x1)2+ (y2 – y1)2] This is simply a variation on c = √(a2+ b2), which is the PythagoreanTheorem if you solve for c2 Pythagorean Triples are sets of 3 numbers that fit the criteria of a2+b2= c2 Since any set of Pythagorean Triples can be multiplied by an infiniteamount of constants, there are an infinite amount of PythagoreanTriples If triangles with side lengths that corresponded to every PrimitivePythagorean Triple (reduced by greatest common factor) were drawn ona Cartesian Plane, we would end up with a unit circle, which is whereour Trigonometric functions come from