Spoken text by ms Jane

voice: so, in general given two fixed sidelengths, lets say this is leg a and this is leg b for a triangle, there are infinite possibilities for the third sidelength, the length of c.

to visualize this, we could for example keep the length of leg a and the length of leg b as they are and simply rotate these two lines around their intersection point, that way c will change its length and leg a and leg b keep apart from each other like this, then c will continuosuly grow, we can also rotate these sides towards each other like this…, then c will continuously shrink, so as said, for a triangle with given lengths of leg a and leg b, there are infinitely many possible combinations for c

however an important prerequisite for the pythagorean theorem was that we are looking at rectangular triangles and as you can see if we constrain the angle between these sides to be rectangular, there is only one possibility for the lenght of c. Given a triangle with fixed length of leg a and leg b, and a rectangular angle between these two sides, the only degree of freedom we could still modify, whilst keeping all neccessary constraints, would be the orientation of the whole triangle, however the orientation doesnt change the length of leg c. This is visible if we rotate our triangle, you see, c doesnt change its length.

So, given a rectangular triangle of which we know two sidelengths, here length of leg a and leg b, there is only one possibility for the third sidelength, here hypotenuse c, because If we made c longer like this whilst keeping the length of leg a and leg b we would no longer have a triangle, and if we kept the lines jointed as a triangle and then increase the length of c you can see that we would break the rectangularity constraint. So, to sum it up, if we keep legs a and b fixed, and we don’t break the triangle or the right-angle constraint, there is only one possible length for the hypotenuse c to turn the 3 lines, into a connected rectangular triangle. The one length, we find using the pythagorean theorem.