Yes, the Pythagorean theorem gives the exact measurements always. It can be backed up by proofs and sin, cosine, etc.
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the measurement is not always exact and say you were to measure a football field and your measurement were to be exact it would be close to impossible
Yes, he must have proved his own Theorem otherwise it would not have been adopted by mathematicians across the globe. I'm sure you could test out the theorem: check whether c2 really does equal b2 + a2 in a manual measurement of a triangle; though this is less accurate and not as precise as the Theorem.
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Since the rectangle has right angles, you can use Pythagoras' Theorem in this case.
Using Pythagoras' theorem the length of the hypotenuse is 39 units of measurement.