Explanation of proof on construction: How to square a rectangle?

Question

By squaring a rectangle, I mean ... "GIven a rectangle, construct a square with the same area of that rectangle"

I found the steps and proof on how to square a rectangle on this site: http://mathworld.wolfram.com/RectangleSquaring.html

However, I do not understand the how they got from the first to the second step of the proof. What makes the area of the rectangle equal to (a+b)(a-b)? I understand the (a+b) side, but what makes the other side (a-b)?

Answers appreciated. Thanks!

Answer 1

a-b=ED=EF

because G is the midpoint of BF.

You neglected to mention that you are constructing a quadrature using just compass and straight-edge.

Answer 2

Squaring A Rectangle

Answer 3

First off, notice that GF is equal in length to the radius of the semicircle. Because we know that the radius of the circle is equal to a, we can say that: GF = a.

Now look back at the diagram. Notice that EF is equal in length to: GF - GE.

We know that the length of GF is a, and we also know that the length of GE is b, so:

EF = a - b

a + b = BE so:

BE * EF = (a + b)(a - b)

Factor that out to get: a^2 - b^2

Using the Pythagorean Theorem you can show that:

b^2 + c^2 = a^2. You could reorder the variables showing that:

a^2 - b^2 = c^2 (the square has the area of c^2).

This proves that the square has the same area as the rectangle.

Hope this helps!

Answer 4

BG=GF because G was the midpoint of BF. EF=GF-GE =BG-GE = a-b by the way lengths a and b are labelled. I thought "squaring a rectangle" was a harder construction but this is only afew steps! The number of steps I remeber from geometry, it was something like 7 steps................

Answer 5

x^2 = 25*18 x = 15sqrt(2)