Name: Alexis Lockwood
Who found the length ratios of: x, 2x, and x root 3 for a 30-60-90 degree triangle? and x, x, x root 2 for a 45-45-90 degree triangle?
Also, how were these ratios discovered, and is there any other significant, interesting information regarding them or their use?
Facts about similar triangles were known by the Greeks, certainly as far back as Pythagoras (Born: about 569 BC in Samos, Ionia Died: about 475 BC). The side length ratios as you state them with irrational numbers (root 2 and root 3) were not known to Pythagoras, or at least not publically stated as Pythagoras refused to accept the existence of the irrational numbers. Some people believe that that one of his students, Hippasus, was sentenced to death for his refusal to deny their existence. It is said that Theodorus (circa 400 BC), Plato's teacher, proved that root 2 ,root 3, root 5 ... are irrational.
The angle measurements for these triangles in degrees comes from the Babylonians. We had an earlier question which asked why a circle is divided into 360o. You might enjoy reading Chris' answer.
Two excellent places to look for material of this kind are the The MacTutor History of Mathematics archive in particluar Babylonian and Egyptian mathematics, and the Earliest Known Uses of Some of the Words of MathematicsCheers,