2018 AMC 10B Problem 22

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?

Show solution

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.

Tagged: geometric probability · law of cosines