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 .
[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 .
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.
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.