2001 AMC 12 Problems/Problem 17
Problem
A point is selected at random from the interior of the pentagon with vertices
,
,
,
, and
. What is the probability that
is obtuse?
Solution
The angle is obtuse if and only if
lies inside the circle with diameter
. (This follows for example from the fact that the inscribed angle is half of the central angle for the same arc.)
The area of is
, and the area of
is
.
From the Pythagorean theorem the length of is
, thus the radius of the circle is
, and the area of the half-circle that is inside
is
.
Therefore the probability that is obtuse is
. Answer choice
See Also
2001 AMC 12 (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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