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 04:59, 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?
See Also
Solution
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.