Problem:
Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $[0,1]$. Which of the following numbers is closest to the probability that $x,y,$ and $1$ are the side lengths of an obtuse triangle?
$\textbf{(A)} \text{ 0.21} \qquad \textbf{(B)} \text{ 0.25} \qquad \textbf{(C)} \text{ 0.29} \qquad \textbf{(D)} \text{ 0.50} \qquad \textbf{(E)} \text{ 0.79}$
Solution:
Note that in order for it to be a triangle, we must have $$x+y>1$$
In order for it to be obtuse, we must have $$1^2 > x^2+y^2$$
Graphing these two inequalities, we see that the total possible area is $1 \cdot 1 = 1$, and the wanted region (the intersection of the two inequalities) is $\frac{\pi}{4} - 0.5$.
Hence, the probability is $\frac{\frac{\pi}{4} - 0.5}{1}$, which is approximately $0.29 \longrightarrow \boxed{\textbf{(C)}}$.
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