Difference between revisions of "2011 AMC 10B Problems/Problem 16"
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== See Also== | == See Also== | ||
− | + | == Solution == | |
− | {{ | + | |
+ | <center><asy> | ||
+ | unitsize(10mm); | ||
+ | defaultpen(linewidth(.8pt)+fontsize(10pt)); | ||
+ | dotfactor=1; | ||
+ | |||
+ | pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)), F=(1+sqrt(2),2+sqrt(2)), G=(1,2+sqrt(2)), H=(0,1+sqrt(2)); | ||
+ | pair I=(1,1), J=(1+sqrt(2),1), K=(1+sqrt(2),1+sqrt(2)), L=(1,1+sqrt(2)); | ||
+ | |||
+ | draw(A--B--C--D--E--F--G--H--cycle); | ||
+ | draw(A--D); | ||
+ | draw(B--G); | ||
+ | draw(C--F); | ||
+ | draw(E--H); | ||
+ | |||
+ | pair[] ps={A,B,C,D,E,F,G,H,I,J,K,L}; | ||
+ | dot(ps); | ||
+ | label("$A$",A,W); | ||
+ | label("$B$",B,S); | ||
+ | label("$C$",C,S); | ||
+ | label("$D$",D,E); | ||
+ | label("$E$",E,E); | ||
+ | label("$F$",F,N); | ||
+ | label("$G$",G,N); | ||
+ | label("$H$",H,W); | ||
+ | label("$I$",I,NE); | ||
+ | label("$J$",J,NW); | ||
+ | label("$K$",K,SW); | ||
+ | label("$L$",L,SE); | ||
+ | label("$\sqrt{2}$",midpoint(B--C),S); | ||
+ | label("$1$",midpoint(A--I),N); | ||
+ | </asy> | ||
+ | </center> | ||
+ | |||
+ | If the side lengths of the dart board and the side lengths of the center square are all <math>\sqrt{2},</math> then the side length of the legs of the triangles are <math>1</math>. | ||
+ | |||
+ | <cmath>\begin{align*} | ||
+ | \text{area of center square} &: \sqrt{2} \times \sqrt{2} = 2\\ | ||
+ | \text{total area} &: (\sqrt{2})^2 + 4(1 \times \sqrt{2}) + 4(1 \times 1 \times \frac{1}{2}) = 2 + 4\sqrt{2} + 2 = 4 + 4\sqrt{2} | ||
+ | \end{align*}</cmath> | ||
+ | |||
+ | Use [[Geometric probability]] by putting the area of the desired region over the area of the entire region. | ||
+ | |||
+ | <cmath> \frac{2}{4+4\sqrt{2}} = \frac{1}{2+2\sqrt{2}} \times \frac{2-2\sqrt{2}}{2-2\sqrt{2}} = \frac{2-2\sqrt{2}}{-4} = \boxed{\textbf{(A)} \frac{\sqrt{2}-1}{2}}</cmath> |
Revision as of 05:59, 6 April 2019
Problem
A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is probability that the dart lands within the center square?
![[asy] unitsize(10mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)), F=(1+sqrt(2),2+sqrt(2)), G=(1,2+sqrt(2)), H=(0,1+sqrt(2)); draw(A--B--C--D--E--F--G--H--cycle); draw(A--D); draw(B--G); draw(C--F); draw(E--H); [/asy]](http://latex.artofproblemsolving.com/6/8/1/68141b2184d1682ae8f8d03c273456191e2cb1bd.png)
See Also
Solution
![[asy] unitsize(10mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=1; pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)), F=(1+sqrt(2),2+sqrt(2)), G=(1,2+sqrt(2)), H=(0,1+sqrt(2)); pair I=(1,1), J=(1+sqrt(2),1), K=(1+sqrt(2),1+sqrt(2)), L=(1,1+sqrt(2)); draw(A--B--C--D--E--F--G--H--cycle); draw(A--D); draw(B--G); draw(C--F); draw(E--H); pair[] ps={A,B,C,D,E,F,G,H,I,J,K,L}; dot(ps); label("$A$",A,W); label("$B$",B,S); label("$C$",C,S); label("$D$",D,E); label("$E$",E,E); label("$F$",F,N); label("$G$",G,N); label("$H$",H,W); label("$I$",I,NE); label("$J$",J,NW); label("$K$",K,SW); label("$L$",L,SE); label("$\sqrt{2}$",midpoint(B--C),S); label("$1$",midpoint(A--I),N); [/asy]](http://latex.artofproblemsolving.com/d/c/4/dc4877d213827aa8426c61a0f2dec96c9f470de2.png)
If the side lengths of the dart board and the side lengths of the center square are all then the side length of the legs of the triangles are
.
Use Geometric probability by putting the area of the desired region over the area of the entire region.