Help understanding square root of two?

Question

In Pythagoras's theorem I know that if a triangle has two sides equal to 1 the hypotenuse must be the square root of two. But my teacher brought up an interesting point the other day, if you kept dividing the triangle up why wouldn't the hypotenuse be 2? I know this has something to do with limits, but can someone please help me? Thanks.

Answer 1

Not sure what you mean by "dividing the triangle up". Suppose the triangle is ABC with AB = hypotenuse and AC and BC are the two equal legs. If AC = BC = 1 then AB = sqrt(2). Now let's divide the triangle into two triangles by drawing a line fro C _|_ AB at D.

This forms triangle ADC whose legs are each sqrt(2)/2, so hypotenuse AC = 1. As you further divide the original triangle into smaller triangles, the hypotenuse becomes smaller and smaller. It could never be 2.

Answer 2

Hypotenuse can be 2 if both of the other sides are equal to √2.