+1252 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
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 \(1:1:\sqrt{2}\)
