Pythagorean Theorem
Pythagorean Theorem
Diana Lorance
MAT126
Dan Urbanski
March 3, 2013
Pythagorean Theorem
In this paper we are going to look at a problem that can be seen in the “Projects” section on page 620 of the Math in our World text. The problem discusses Pythagorean triples and asks if you can find more Pythagorean triples than the two that are listed which are (3,4, and 5) and (5,12, and 13) (Bluman, 2012). The Pythagorean theorem states that for any right triangle, the sum of the squares of the length of the sides of the triangle is equal to the square of the length of the side opposite of the right angle (hypotenuse) and can be shown as a² + b² = c² (Bluman, 2012). We will be using a formula to find five more Pythagorean Triples and then verify each of them in the Pythagorean Theorem equation.
The formula that I have decided to illustrate is (2m)² + (m2 - 1)² = (m2 + 1)² where m is any natural number, this formula is attributed to Plato (c. 380 B. C.) (Edenfield, 1997). A natural number is any number starting from one that is not a fraction or negative (MathIsFun, 2011). The triples will be the square roots of each part of the equation. We will test this formula with the natural numbers of 5, 8, and 10. When we use 5 the formula looks like this: (2x5)² + (5² - 1)² = (5² + 1)², 10² + (25 – 1)² = (25 + 1)², 100 + 24² = 26², 100 + 576 = 676. After taking the square roots of each we have 10, 24, 26 for the triples of 5. When we use 8 the formula looks like this: (2x8)² + (8² - 1)² = (8² + 1)², 16² + (64 – 1)² = (64 + 1)², 256 + 63² = 65², 256 + 3969 = 4225, therefore; the triples for 8 are 16, 63, 65. When we use 10 the formula looks like this: (2x10)² + (10² - 1)² = (10² + 1)², 20² + (100 – 1)² = (100 + 1)², 400 + 99² = 101², 400 + 9801 = 10201, therefore; the triples for 10 are 20, 99, 101. Now that we have established the triples of (10, 24, 26), (16, 63, 65), and (20, 99, 101) we can check them using the Pythagorean Theorem equation.
The Pythagorean
Diana Lorance
MAT126
Dan Urbanski
March 3, 2013
Pythagorean Theorem
In this paper we are going to look at a problem that can be seen in the “Projects” section on page 620 of the Math in our World text. The problem discusses Pythagorean triples and asks if you can find more Pythagorean triples than the two that are listed which are (3,4, and 5) and (5,12, and 13) (Bluman, 2012). The Pythagorean theorem states that for any right triangle, the sum of the squares of the length of the sides of the triangle is equal to the square of the length of the side opposite of the right angle (hypotenuse) and can be shown as a² + b² = c² (Bluman, 2012). We will be using a formula to find five more Pythagorean Triples and then verify each of them in the Pythagorean Theorem equation.
The formula that I have decided to illustrate is (2m)² + (m2 - 1)² = (m2 + 1)² where m is any natural number, this formula is attributed to Plato (c. 380 B. C.) (Edenfield, 1997). A natural number is any number starting from one that is not a fraction or negative (MathIsFun, 2011). The triples will be the square roots of each part of the equation. We will test this formula with the natural numbers of 5, 8, and 10. When we use 5 the formula looks like this: (2x5)² + (5² - 1)² = (5² + 1)², 10² + (25 – 1)² = (25 + 1)², 100 + 24² = 26², 100 + 576 = 676. After taking the square roots of each we have 10, 24, 26 for the triples of 5. When we use 8 the formula looks like this: (2x8)² + (8² - 1)² = (8² + 1)², 16² + (64 – 1)² = (64 + 1)², 256 + 63² = 65², 256 + 3969 = 4225, therefore; the triples for 8 are 16, 63, 65. When we use 10 the formula looks like this: (2x10)² + (10² - 1)² = (10² + 1)², 20² + (100 – 1)² = (100 + 1)², 400 + 99² = 101², 400 + 9801 = 10201, therefore; the triples for 10 are 20, 99, 101. Now that we have established the triples of (10, 24, 26), (16, 63, 65), and (20, 99, 101) we can check them using the Pythagorean Theorem equation.
The Pythagorean
References: Bluman, Allen G. (2012). MAT!26 Vol.1 Math in our World. Edenfield, Kelly (1997). Pythagorean Triples. Retrieved from http://jwilson.coe.uga.edu/EMT668/EMT668.Folders.F97/Edenfield/Pythtriples/Pythripl s.html MathIsFun (2011). Natural Number. Math is Fun Dictionary.