Difference between revisions of "2009 AMC 12B Problems/Problem 23"
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(Alternately, when we got to the point that we know that a single red triangle is <math>\frac 29</math>, we can directly note that the picture is symmetric, hence we can just consider the first quadrant and there the probability is <math>1 - \frac 29 = \frac 79</math>. This saves us the work of first multiplying and then dividing by <math>4</math>.) | (Alternately, when we got to the point that we know that a single red triangle is <math>\frac 29</math>, we can directly note that the picture is symmetric, hence we can just consider the first quadrant and there the probability is <math>1 - \frac 29 = \frac 79</math>. This saves us the work of first multiplying and then dividing by <math>4</math>.) | ||
+ | == Solution 2 (Same idea) == | ||
+ | The solution proposed above is good, but there is a more straightforward method. First, turn <math>\frac34 + \frac34i</math> into polar form as <math>\frac{3\sqrt{2}}{4}e^{\frac{\pi}{4}i}</math>. Restated using geometric probabilities, we are trying to find the portion of a square enlarged by a factor of <math>\frac{3\sqrt{2}}{4}</math> and rotated <math>45\degree</math> | ||
+ | |||
+ | <asy> | ||
+ | unitsize(2cm); | ||
+ | defaultpen(0.8); | ||
+ | path s = (-1,-1) -- (-1,1) -- (1,1) -- (1,-1) -- cycle; | ||
+ | path t = (4/3,0) -- (0,4/3) -- (-4/3,0) -- (0,-4/3) -- cycle; | ||
+ | path s_cap_t = (1/3,1) -- (1,1/3) -- (1,-1/3) -- (1/3,-1) -- (-1/3,-1) -- (-1,-1/3) -- (-1,1/3) -- (-1/3,1) -- cycle; | ||
+ | filldraw(s, lightred, black); | ||
+ | filldraw(t, lightgreen, black); | ||
+ | filldraw(s_cap_t, lightyellow, black); | ||
+ | draw( (-5/3,0) -- (5/3,0), dashed ); | ||
+ | draw( (0,-5/3) -- (0,5/3), dashed ); | ||
+ | </asy> | ||
== See Also == | == See Also == |
Revision as of 20:32, 12 December 2018
Problem
A region in the complex plane is defined by
A complex number
is chosen uniformly at random from
. What is the probability that
is also in
?
Solution
We can directly compute .
This number is in if and only if
and at the same time
. This simplifies to
and
.
Let , and let
denote the area of the region
. Then obviously the probability we seek is
. All we need to do is to compute the area of the intersection of
and
. It is easiest to do this graphically:
Coordinate axes are dashed, is shown in red,
in green and their intersection is yellow. The intersections of the boundary of
and
are obviously at
and at
.
Hence each of the four red triangles is an isosceles right triangle with legs long , and hence the area of a single red triangle is
. Then the area of all four is
, and therefore the area of
is
. Then the probability we seek is
.
(Alternately, when we got to the point that we know that a single red triangle is , we can directly note that the picture is symmetric, hence we can just consider the first quadrant and there the probability is
. This saves us the work of first multiplying and then dividing by
.)
Solution 2 (Same idea)
The solution proposed above is good, but there is a more straightforward method. First, turn into polar form as
. Restated using geometric probabilities, we are trying to find the portion of a square enlarged by a factor of
and rotated $45\degree$ (Error compiling LaTeX. Unknown error_msg)
See Also
2009 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
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.