2018 AMC 10B #22

 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?

$\mathbf{\text{(A)}}\text{ 0.21}\mathbf{\text{(B)}}\text{ 0.25}\mathbf{\text{(C)}}\text{ 0.29}\mathbf{\text{(D)}}\text{ 0.50}\mathbf{\text{(E)}}\text{ 0.79}$

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 $0.29⟶\boxed{{\displaystyle \mathbf{\text{(C)}}}}$.