Hello, GAdams!

I have a rectangle sides of $\displaystyle \sqrt{2}$ and $\displaystyle 1.$

I cut it in half to show that the ratio of the new rectangles is the same as the original.

Here's what I did:

New rectangle is $\displaystyle \frac{\sqrt{2}}{2}\,:\,1$ . . . . not quite correct

This is the original rectangle: Code:

* - - - - - - - - - *
| |
| |
1 | | 1
| |
| |
* - - - - - - - - - *
√2

The ratio of length-to-width is: .$\displaystyle {\color{blue}\sqrt{2}\,:\,1}$

Bisect it: Code:

* - - - - * - - - - *
| | |
| | |
1 | | | 1
| | |
| | |
* - - - - * - - - - *
½√2 ½√2

Each small rectangle looks like this: Code:

* - - - - - *
| |
| | ½√2
| |
* - - - - - *
1

The ratio of length-to-width is: .$\displaystyle {\color{blue}1\,:\,\frac{\sqrt{2}}{2}}$

And we must show that the two ratios are equal.

The first ratio is: .$\displaystyle \frac{\sqrt{2}}{1} \;=\;{\color{red}\sqrt{2}}$

The second ratio is: .$\displaystyle \frac{1}{\frac{\sqrt{2}}{2}} \:=\:\frac{2}{\sqrt{2}} $

. . Rationalize: .$\displaystyle \frac{2}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}} \;=\;\frac{2\sqrt{2}}{2} \;=\;{\color{red}\sqrt{2}}$

Therefore, the ratios are equal.