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This is just a fun question:

Prove a 45-45-90 triangle has side ratios \(1:1:\sqrt2\) WITHOUT Pythagorean Theorem or Trigonometry.

Can you do it?

(also, I'm kind of curious because I don't know how to prove it without Pythag. or Trig, is it even possible?)



 Dec 13, 2018

Sure, it's possible by considering the area. We can assume that the legs have a length 1 and so we wish to find the length of the hypothenuse which we will call x. The area then becomes \(\frac{1}{2}\). If we draw the height of the triangle from the hypothenuse we notice that it has the length \(\frac{x}{2} \) and so the area can also be calculated to be \(\frac{x \cdot \frac{x}{2}}{2}\). We then get the equation \(\frac{x^2}{4}=\frac{1}{2} \) or \(x^2=2\) and so \(x= \sqrt{2}\) and the ratio is \(1:1:\sqrt{2}\)

 Dec 13, 2018

Well, if we were dumb, how would we know the height is \(x/2\) ?

CoolStuffYT  Dec 13, 2018

Well that's because when we draw the height we split the triangle into two isosceles triangles and also split the hypothenuse into two equal parts of length  \(x/2\). Because this is one of the legs and the other leg is the height they must have the same length.

Guest Dec 14, 2018

Oh yes, you are right! Nice proof there!

CoolStuffYT  Dec 15, 2018

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