2011 AMC 12B Problems/Problem 12
Problem
A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?
Solution
Let's assume that the side length of the octagon is . The area of the center square is just
. The triangles are all
triangles, with a side length ratio of
. The area of each of the
identical triangles is
, so the total area of all of the triangles is also
. Now, we must find the area of all of the 4 identical rectangles. One of the side lengths is
and the other side length is
, so the area of all of the rectangles is
. The ratio of the area of the square to the area of the octagon is
. Cancelling
from the fraction, the ratio becomes
. Multiplying the numerator and the denominator each by
will cancel out the radical, so the fraction is now
There's a similar but simpler way that what the user above has done. You set one of the sides to be 2, then the area of the middle square is , the combined area of the 4 triangles are
, the combined areas of the rectangles on the side become 8 sqrt2, you add all of that up. Which is 8+8 sqrt2, then 4 divide that is going to give you A
See also
2011 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.