Last week, we have discussed the second part of our Paper Folding series, a fold that extracts the cube root of any number. In this post, we are going to discuss its proof, but before that, let’s recall how to do the paper fold.

**Paper Folding Instructions **

- Get a rectangular piece of paper and fold it in the middle, horizontally and vertically, and let the creases (see green segments in the applet) represent the coordinate axes.
- Let
*M*denote (0,1) and let*R*denote (-*r*,0). - Make a single fold that places
*M*on*y*= -1 and*R*on*x=r*. - The x-intercept of the fold is .

*P*and

*Q*to satisfy the conditions above. Note that you can also move point

*R*.

[iframe http://mathandmultimedia.com/wp-content/uploads/2011/07/paperfoldcuberoot.html 565 432]

**Theorem**: Prove that if *M* on and is on , the intersection of the fold and the *x*-axis is at .

**Proof: **

One possible fold satisfying the conditions above is shown in the figure below. Checking the *Hint check box* reveals three triangles: *MNO*, *POR*, and *PON*.

It is clear that the three triangles are similar (Can you see why?).This implies that

Equating the first and second ratios, we have which means that . Also, equating ratios 1 and 3, we have which results to . This means that giving us . But the length of , therefore and we are done with our proof.