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Difference between revisions of "2014 AMC 10B Problems/Problem 19"

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==Problem==
 
==Problem==
 +
Two concentric circles have radii <math>1</math> and <math>2</math>. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?
 +
 +
<math>
 +
\textbf{(A)}\ \frac{1}{6}\qquad
 +
\textbf{(B)}\ \frac{1}{4}\qquad
 +
\textbf{(C)}\ \frac{2-\sqrt{2}}{2}\qquad
 +
\textbf{(D)}\ \frac{1}{3}\qquad
 +
\textbf{(E)}\ \frac{1}{2}\qquad</math>
 +
[[Category:Introductory Geometry Problems]]
  
 
==Solution==
 
==Solution==
  
−
==See Also==
+
Let the center of the two circles be <math>O</math>. Now pick an arbitrary point <math>A</math> on the boundary of the circle with radius <math>2</math>. We want to find the range of possible places for the second point, <math>A'</math>, such that <math>AA'</math> passes through the circle of radius <math>1</math>. To do this, first draw the tangents from <math>A</math> to the circle of radius <math>1</math>. Let the intersection points of the tangents (when extended) with circle of radius <math>2</math> be <math>B</math> and <math>C</math>. Let <math>H</math> be the foot of the altitude from <math>O</math> to <math>\overline{BC}</math>. Then we have the following diagram.
 +
 
 +
<asy>
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scale(200);
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pair A,O,B,C,H;
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A = (0,1);
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O = (0,0);
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B = (-.866,-.5);
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C = (.866,-.5);
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H = (0, -.5);
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draw(A--C--cycle);
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draw(A--O--cycle);
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draw(O--C--cycle);
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draw(O--H,dashed+linewidth(.7));
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draw(A--B--cycle);
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draw(B--C--cycle);
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draw(O--B--cycle);
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dot("$A$",A,N);
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dot("$O$",O,NW);
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dot("$B$",B,W);
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dot("$C$",C,E);
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dot("$H$",H,S);
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label("$2$",O--(-.7,-.385),N);
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label("$1$",O--H,E);
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draw(circle(O,.5));
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draw(circle(O,1));
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</asy>
 +
 
 +
We want to find <math>\angle BOC</math>, as the range of desired points <math>A'</math> is the set of points on minor arc <math>\overarc{BC}</math>. This is because <math>B</math> and <math>C</math> are part of the tangents, which "set the boundaries" for <math>A'</math>. Since <math>OH = 1</math> and <math>OB = 2</math> as shown in the diagram, <math>\triangle OHB</math> is a <math>30-60-90</math> triangle with <math>\angle BOH = 60^\circ</math>. Thus, <math>\angle BOC = 120^\circ</math>, and the probability <math>A'</math> lies on the minor arc <math>\overarc{BC}</math> is thus <math>\dfrac{120}{360} = \boxed{\textbf{(D)}\: \dfrac13}</math>.
 +
 
 +
==Comment==
 +
This problem is exactly the Bertrand paradox! I feel sorry for students who took 2014 AMC 10B :-(
 +
The answers (E) 1/2, (D) 1/3, and (B) 1/4 are all correct, depending on what kind of uniform you choose.
 +
(1) If you choose angle uniform, the probability is 1/3.
 +
(2) If you choose the center of the chord is uniformly distributed in the big circle, the probability is 1/4.
 +
(3) If you choose the center of the chord is uniformly distributed along a radius (i.e. 0 to 2 from the center of the big circle), the probability is 1/2.
 +
 
 +
-EUCLID_2003
 +
 
 
{{AMC10 box|year=2014|ab=B|num-b=18|num-a=20}}
 
{{AMC10 box|year=2014|ab=B|num-b=18|num-a=20}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 09:27, 7 October 2024

Problem

Two concentric circles have radii $1$ and $2$. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?

$\textbf{(A)}\ \frac{1}{6}\qquad \textbf{(B)}\ \frac{1}{4}\qquad \textbf{(C)}\ \frac{2-\sqrt{2}}{2}\qquad \textbf{(D)}\ \frac{1}{3}\qquad \textbf{(E)}\ \frac{1}{2}\qquad$

Solution

Let the center of the two circles be $O$. Now pick an arbitrary point $A$ on the boundary of the circle with radius $2$. We want to find the range of possible places for the second point, $A'$, such that $AA'$ passes through the circle of radius $1$. To do this, first draw the tangents from $A$ to the circle of radius $1$. Let the intersection points of the tangents (when extended) with circle of radius $2$ be $B$ and $C$. Let $H$ be the foot of the altitude from $O$ to $\overline{BC}$. Then we have the following diagram.

[asy] scale(200); pair A,O,B,C,H; A = (0,1); O = (0,0); B = (-.866,-.5); C = (.866,-.5); H = (0, -.5); draw(A--C--cycle); draw(A--O--cycle); draw(O--C--cycle); draw(O--H,dashed+linewidth(.7)); draw(A--B--cycle); draw(B--C--cycle); draw(O--B--cycle); dot("$A$",A,N); dot("$O$",O,NW); dot("$B$",B,W); dot("$C$",C,E); dot("$H$",H,S); label("$2$",O--(-.7,-.385),N); label("$1$",O--H,E); draw(circle(O,.5)); draw(circle(O,1)); [/asy]

We want to find $\angle BOC$, as the range of desired points $A'$ is the set of points on minor arc $\overarc{BC}$. This is because $B$ and $C$ are part of the tangents, which "set the boundaries" for $A'$. Since $OH = 1$ and $OB = 2$ as shown in the diagram, $\triangle OHB$ is a $30-60-90$ triangle with $\angle BOH = 60^\circ$. Thus, $\angle BOC = 120^\circ$, and the probability $A'$ lies on the minor arc $\overarc{BC}$ is thus $\dfrac{120}{360} = \boxed{\textbf{(D)}\: \dfrac13}$.

Comment

This problem is exactly the Bertrand paradox! I feel sorry for students who took 2014 AMC 10B :-( The answers (E) 1/2, (D) 1/3, and (B) 1/4 are all correct, depending on what kind of uniform you choose. (1) If you choose angle uniform, the probability is 1/3. (2) If you choose the center of the chord is uniformly distributed in the big circle, the probability is 1/4. (3) If you choose the center of the chord is uniformly distributed along a radius (i.e. 0 to 2 from the center of the big circle), the probability is 1/2.

-EUCLID_2003

2014 AMC 10B (Problems • Answer Key • Resources)
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Problem 18 		Followed by
Problem 20
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