On old tablets the Babylonians were able to work out the digits to the square root of two from the hypotenuse of a $45^\circ-45^\circ-90^\circ$ triangle.
How could they have figured this out without the use of Pythagorean theorem unless they came up with it first?
On
There is a cuneiform tablet in the British Museum, BM 85196, which is dated to the Old Babylonian period (2000 BCE - 1600 BCE). Problem 9 on that tablet is understood as being about a pole, initially vertical, sliding out from a wall so that the pole forms the hypotenuse of a right triangle and the wall and the ground the legs. Note that the wall is not explicitly mentioned in the problem, although it is possible that the damaged part contained such a mention. The following description, with translation due to Jens Høyrup, can be found in the article
Duncan J. Melville, Poles and walls in Mesopotamia and Egypt, Historia Mathematica, Volume 31, Issue 2, 2004,
which discusses a number of similar problems found across a variety of cultures and time periods.
Small portions of Problem 9 are missing or damaged, including, unfortunately, a part of the first line, but the structure of the problem can be safely restored. The problem reads as follows:
A pole. 30 (a reed). From its …
Above, it has descended 6, below, how far has it moved away?
You. Square 30: 15 you see. Subtract 6 from 30: 24 you see.
Square 24: 9,36 you see. Subtract 9,36 from 15:
5,24 you see. What is the square root of 5,24? 18 is the square root. 18
on the ground it has moved away. If 18 on the ground
above, what did it descend? Square 18: 5,24 you see.
Subtract 5,24 from 15: 9,36 you see. Of 9,36,
what is the square root? 24 is the square root. Subtract 24 from 30:
6 you see descended. Such is the procedure.
In this passage, base-60 numbers are given with digits separated by commas. As in the original there is no explicit sexagesimal point, and the magnitude of the numbers must be inferred from context. So $30$ means $30/60=1/2$, $6$ means $6/60=1/10$, $15$ means $15/60=1/4$, $9,36$ means $9/60+36/60^2=576/3600=4/25$, and so on. Following the procedure, the length along the ground is computed, in modern terms, as $$ \sqrt{\text{(length of pole)}^2-\text{(height on wall)}^2}=\sqrt{(1/2)^2-(1/2-1/10)^2}=3/10 $$
Computing the hypotenuse of the unit right triangle (same as the diagonal of the unit square) is significantly easier than the general Pythagorean Theorem, see, e.g., this
For that matter, the so-called Babylonian method for extracting square roots doesn't rely on the Pythagorean Theorem.