Proofs of Pythagoras’s theorem by shearing, by similar triangles, and by “moving triangles”

Pythagoras by shearing

That is a proof of Pythagoras’s theorem by shearing (which we came across in FP1: remember? Remember the Mona Lisa picture in 2B6 which illustrates shearing?)



On 8 January 2015 Serafina Cuomo talked about the different ways we can see Pythagoras’s theorem. Here is another proof which uses similar triangles, and doesn’t draw the squares at all.

proof51

ABC is the right-angled triangle, and AD is the height from A to BC. The triangles ABC, DBA, and DAC are similar which leads to two ratios:

AB/BC = BD/AB and AC/BC = DC/AC

Written another way these become

AB·AB = BD·BC and AC·AC = DC·BC

Summing up we get

AB·AB + AC·AC = BD·BC + DC·BC = BC·BC



Another proof, and perhaps the easiest, is from the diagram below.

pythag-move-triangles

There are many hundreds of other proofs: see Proofs of Pythagoras.