User:MartinPalanca

Martin Palanca, 1st Year Faculty of Forestry

HOMEWORK, The Pythagorean Theorem

The Pythagorean theorem, developed by the Greek scholar named Pythagoras, is used to determine an unknown side of a right triangle given the length of the two other sides. This theorem is represented by the formula a^2 + b^2 = c^2. This relation explains that the length of the hypotenuse, c, is equal to the square root of the sum of the squares of the other two sides. Ex. c = (b^2 + a^2)^1/2. Alternatively, given the length of the hypotenuse and one other side, the unknown side can be determined as follows: a = (c^2 –b^2)^1/2 or b = (c^2 – a^2)^1/2.

Example: David is runs 5m and then comes to a stop. He then throws a ball straight up into the air. The ball reaches a maximum height of 6m. How far is the maximum height of the ball from David’s original starting position?

Given: A= 5m, B= 6m, C = ? m

SOLUTION

A^2 + b^2 = c^2

5^2 + 6^2 = c^2 (plug in known values)

25+36 = c^2

(61)^1/2 = c (to isolate c, find the square root of both sides)

c = 7.81m

Therefore, the distance between the ball's maximum height and David's original starting position is 7.81m

To simplify problems regarding the Pythagorean theorem, the discovery and use of the Pythagorean triples has proven very efficient. These triples are predetermined values that satisfy the a^2 + b^2 = c^2 formula. Some examples of these triples, in (a,b,c) format, are (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17). In these triples, the square root of the sum of the squares of the first two numbers always equals the third.

Many formulas have been developed based on the Pythagorean theorem. One in particular, which is also widely used in the world of trigonometry, is the Cosine Law. This law allows the determination of the length of one side of any triangle (whether a right triangle or not) given the two other sides and the angle between them: a^2 + b^2 - 2ab cos(C) = c^2

The Pythagorean theorem has been very important to the evolution of mathematics and the field of trigonometry. Its use has become very important in mathematical calculations in everyday life.

SOURCES: http://mathematica.ludibunda.ch/pythagoras6.html http://www.mathsisfun.com/algebra/trig-cosine-law.html http://en.wikipedia.org/wiki/Pythagorean_theorem http://en.wikipedia.org/wiki/Law_of_cosines