2015 HMMT November General Round Problem 2

Let a and b be real numbers randomly (and independently) chosen from the range [0,1]. Find the probability that a, b and 1 form the side lengths of an obtuse triangle.

Answers are checked by value, so any equivalent form is accepted: 1/2, \frac{1}{2} and 0.5 all count as the same answer.

Full credit goes to HMMT for authoring these problems. This problem is from the November 2015 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.