Difference between revisions of "2005 AMC 12B Problems/Problem 18"
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Revision as of 13:54, 4 July 2011
Problem
Let 
and 
be points in the plane. Define 
as the region in the first quadrant consisting of those points 
such that 
is an acute triangle. What is the closest integer to the area of the region ?
Solution
For angle 
and 
to be acute, must be between the two lines that are perpendicular to 
and contain points and . For angle to be acute, first draw a 
triangle with as the hypotenuse. Note cannot be inside this triangle's circumscribed circle or else 
. Hence, the area of is the area of the large triangle minus the area of the small triangle minus the area of the circle, which is 
, which is approximately 
. The answer is 
.
See also
| 2005 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 17 |
Followed by Problem 19 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
