User:ShannonLee

Hello, I'm a third year Political Science student, who is planning on doing a secondary degree in Environmental Sciences. I haven't done math in quite awhile so I'm quite rusty at it, hopefully this course will enable me to catch up and improve upon my mathematical skills!

http://wiki.ubc.ca/Politics%26math

The Pythagorean Theorem

 $a^{2} + b^{2} = c^{2}$

Image obtained from [1]

The Pythagorean Theorem is used to solve for the hypotenuse or length of a side of a right angle triangle. The Pythagorean equation, which is $a^{2} + b^{2} = c^{2}$ , was created by Pythagoras, an ancient Greek mathematician; however, some argue that the theorem was realized centuries prior to Pythagoras by the Babylonians.

In the equation $a^{2} + b^{2} = c^{2}$ , a and b refer to the lengths of the triangle, and c refers to the hypotenuse. Rearranging the equation to form $b = \sqrt{(c^{2} - a^{2})}$ or $a = \sqrt{(c^{2} + b^{2})}$ solves for the length of a side of the triangle given the length of one other side and the hypotenuse. Because the theorem is a statement of both areas and lengths, it is said to be an expression of both algebra and geometry.

There are many proofs which justify the Pythagorean Theorem, the following is a proof that was constructed by U.S. president James Garfield. The proof uses a trapezoid to prove that in regards to a right triangle, the sum of the squares of the sides is equal to the square of the hypotenuse. Ultimately this proof uses the formula $\frac{(a + b) (a + b)}{2} = a b + \frac{(c^{2})}{2}$ , with the left side expanded to $a^{2} + 2 a b + b^{2}$ to prove the equality found thousands of years previous by Pythagoras.

Image obtained from: [2]

An example using the Pythagorean Theorem to find the hypotenuse of a right triangle: Joan lives 8.0m east of her high school, Bill lives 10.0m north from the same school. How far do Joan and Bill live from one another?

To solve this question, let a = Joan’s distance from the school (8.0m) and let b= Bill’s distance from the school (10.0m). The Pythagorean Theorem states that $a^{2} + b^{2} = c^{2}$ , Therefore, $8.0 m^{2} + 10.0 m^{2} = c^{2}$ ,

$164.0 m = c^{2}$ $c = \sqrt{164.0}$ $c = 12.8 m$

Sources:

Oracle Education Foundation. (August 25, 2010). The Pythagorean Theorem. Retrieved from: http://library.thinkquest.org/25672/newpage1.htm

Page, John. (2009). Pythagorean Theorem. Retrieved from: http://www.mathopenref.com/pythagorastheorem.html