College Algebra Chapter 1/Section 3 Prove the Pythagorean Theorem. Explain each step along the way.
The Pythagorean Theorem says that, in a right triangle, the a^2 + b^2 = c^2 Proof: We can show that a^2 + b^2 = c^2 using Algebra Take a look at this diagram: it has that "abc" triangle in it (four of them actually): It is a big square, with each side having a length of a+b, so the total area is: A = (a+b)(a+b) Now let's add up the areas of all the smaller pieces: First, the smaller (tilted) square has an area of: c^2 Each of the four triangles has an area of: ab/2 So all four of them together is: 4ab/2 = 2ab Adding up the tilted square and the 4 triangles gives: A = c^2 + 2ab The area of the large square is equal to the area of the tilted square and the 4 triangles. This can be written as: (a+b)(a+b) = c^2 + 2ab NOW, let us rearrange this to see if we can get the Pythagoras theorem: Start with: (a+b)(a+b) = c2 + 2ab Expand (a+b)(a+b): a^2 + 2ab + b^2 = c^2 + 2ab Subtract "2ab" from both sides: a^2 + b^2 = c^2 DONE!
