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# Easy Proof, or is it not?

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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?)

:P

Dec 13, 2018

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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 $$1:1:\sqrt{2}$$

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Dec 13, 2018
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Well, if we were dumb, how would we know the height is $$x/2$$ ?

CoolStuffYT  Dec 13, 2018
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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
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Oh yes, you are right! Nice proof there!

CoolStuffYT  Dec 15, 2018