preview

How Does The Pythagorean Theorem Work?

Decent Essays
Open Document

a2+b2=c2 is the famous theorem that Pythagoras discovered and named, calling it the Pythagorean Theorem. This theorem applies to the right triangle stating, that by adding the length of both legs squared you can then find the squared length of the hypotenuse. This theorem is set up in way that if you know two of the variables, whether it is a leg(b or a) and the hypotenuse (c) or both legs (a and b), you will always be able to find the third measurement. However, why does this theorem work? Why does a2+b2=c2? That is the question that is asked hundreds of times by thousands of people. The answer to it is not a complicated one, the reasoning behind that is because there are at least 367 Pythagorean Theorem proofs out there (Source four). They …show more content…

Step Two: Off of the height, length, and hypotenuse create a coloured square. (Green and blue squares are on the legs and the red square on the hypotenuse).
Step Three: By doing so we acquire the following areas for the square;
Area of the green square is a2
Area of the blue square is b2
Area of the red square is c2
Step Four: If you apply the information that we now have to the pythagorean theorem we would have area of green square + area of blue square = are of red square.
Step Five: If we substitute this with the information we came up with in step three then we have a2+b2=c2.

Proof Two
Step One: Create a random right triangle with the legs the length of a and b, and hypotenuse the length of c.
Step Two: Create an additional copy of the right triangle and position it the same way as the example below. (This creates a trapezoid).
Step Three: Identify the height and length of the trapezoid. (The parallel sides of the trapezoid have a length of a and b. The height of the trapezoid is a+b.)
Step Four: We can now find the area of the trapezoid which is A=12(a+b)(a+b)=12(a+b)2
Step Five: However, since the area of the trapezoid is also the sum of the three triangles that create the trapezoid the area is; …show more content…

Step Seven: You then solve the equation. c2=(a-b)2+2ab =a2-2ab+b2+2ab = a2+b2

Proof Four
Step One: Create four copies of a right triangle
(Note: Each has an area ofab/2)
Step Two: Combine the the triangle so that it forms a square with the side of (a+b) and a hole with the side of c.
Step Three: Then calculate the area of the big square in two ways by creating the equation; (a+b)2=4ab/2 +c2. (When simplified you will get the required identity.)
Step Four: We then use the identity created in the third proof and add it to the identity we created in step three.
Step Five: Add the two identities together. c2=(a-b)2+4ab/2 c2=(a+b)2-4ab/2
2c2=2a2+2b2
c2=a2+b2

Overall there are hundreds of ways to prove the Pythagorean Theorem. As stated before these proofs range from effortless geometric and algebraic proofs to intricate and complex trigonometric and calculus proofs. The proofs shown above are predominantly straightforward and easy to understand and do. Once again, these proofs are only a small portion of other proofs out there in the world. However, it will never matter what proof you choose to prove the Pythagorean Theorem, because no matter what they will all have the same end

Get Access
Related
Get Access