Question

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?

$0.21$

$0.25$

$0.29$

$0.50$

$0.79$

Solution:

The three lengths $x, y, 1$ make a triangle iff $x + y > 1.$ Since $1$ is the longest side, that triangle is obtuse iff $x^2 + y^2 < 1.$ So in the unit square we want the region inside the quarter circle $x^2 + y^2 = 1$ but above the line $x + y = 1.$ That's the quarter disk with the right triangle under the chord removed: $\tfrac{\pi}{4} - \tfrac12 \approx 0.285.$ The closest choice is $0.29.$ Therefore, the answer is C.

2018 AMC 10B Problem 22

Solution:

Problem 22 in Other Years