Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2011 AMC 10B Problems/Problem 16"

(→‎Solution)
m (→‎Solution 2)
 
(13 intermediate revisions by 9 users not shown)
Line 1: Line 1:
 
== Problem==
 
== 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?
+
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 the probability that the dart lands within the center square?
  
 
<center><asy>
 
<center><asy>
Line 21: Line 21:
 
<math> \textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad\textbf{(E)}\ 2 - \sqrt{2}</math>
 
<math> \textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad\textbf{(E)}\ 2 - \sqrt{2}</math>
 
[[Category: Introductory Geometry Problems]]
 
[[Category: Introductory Geometry Problems]]
 +
 +
== 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>
 +
 +
 +
Explanation:
 +
The area of the octagon consists of the area of the triangles, the rectangles, and the square in the middle. Assume the octagon has side length <math>1</math>. The triangles are right isosceles triangles with the hypotenuse 1, so their side length is <math>\frac{\sqrt{2}}{2}</math> and the area of one triangle is <math>\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2}}{2}\cdot\frac{1}{2}=\frac{1}{4}</math>. The area of all 4 triangles is then just <math>1</math>. The rectangles share one side with the octagon, and another with the triangle. The octagon has side length 1, and the triangle has side length <math>\frac{\sqrt{2}}{2}</math> as found earlier. So the area of one rectangle is <math>\frac{\sqrt{2}}{2}</math>. The area of all 4 is <math>4\cdot\frac{\sqrt{2}}{2}=2\sqrt{2}</math>. Finally, the area of the square in the middle is <math>1\cdot1=1</math>. The total area is <math>1+1+2\sqrt{2}=2+2\sqrt{2}</math>. We want the area of the square over the area of the octagon, which is <math>\frac{1}{2+2\sqrt{2}}</math>. Rationalize by multiplying both numerator and denominator by <math>2-2\sqrt{2}</math>: <math>\frac{1}{2+2\sqrt{2}}\cdot\frac{2-2\sqrt{2}}{2-2\sqrt{2}}=\frac{2-2\sqrt{2}}{\left(2+2\sqrt{2}\right)\left(2-2\sqrt{2}\right)}</math>. By the difference of squares, the denominator reduces to <math>-4</math> and the fraction is <math>\frac{2-2\sqrt{2}}{-4}=\frac{\sqrt{2}-1}{2}</math> which is <math>\boxed{\textbf{(A) } \frac{\sqrt{2}-1}{2}}</math>.
 +
 +
~Explanation by JH. L
 +
 +
== Solution 2==
 +
The area of a regular octagon is <math>2(1+\sqrt{2})a^2</math> where <math>a</math> is the side. Hence the answer is obvious now.
  
 
== See Also==
 
== See Also==

Latest revision as of 17:54, 2 September 2024

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 the 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]

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

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]

If the side lengths of the dart board and the side lengths of the center square are all $\sqrt{2},$ then the side length of the legs of the triangles are $1$.

\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*}

Use Geometric probability by putting the area of the desired region over the area of the entire region.

\[\frac{2}{4+4\sqrt{2}} = \frac{1}{2+2\sqrt{2}} \times \frac{2-2\sqrt{2}}{2-2\sqrt{2}} = \frac{2-2\sqrt{2}}{-4} = \boxed{\textbf{(A)} \frac{\sqrt{2}-1}{2}}\]


Explanation: The area of the octagon consists of the area of the triangles, the rectangles, and the square in the middle. Assume the octagon has side length $1$. The triangles are right isosceles triangles with the hypotenuse 1, so their side length is $\frac{\sqrt{2}}{2}$ and the area of one triangle is $\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2}}{2}\cdot\frac{1}{2}=\frac{1}{4}$. The area of all 4 triangles is then just $1$. The rectangles share one side with the octagon, and another with the triangle. The octagon has side length 1, and the triangle has side length $\frac{\sqrt{2}}{2}$ as found earlier. So the area of one rectangle is $\frac{\sqrt{2}}{2}$. The area of all 4 is $4\cdot\frac{\sqrt{2}}{2}=2\sqrt{2}$. Finally, the area of the square in the middle is $1\cdot1=1$. The total area is $1+1+2\sqrt{2}=2+2\sqrt{2}$. We want the area of the square over the area of the octagon, which is $\frac{1}{2+2\sqrt{2}}$. Rationalize by multiplying both numerator and denominator by $2-2\sqrt{2}$: $\frac{1}{2+2\sqrt{2}}\cdot\frac{2-2\sqrt{2}}{2-2\sqrt{2}}=\frac{2-2\sqrt{2}}{\left(2+2\sqrt{2}\right)\left(2-2\sqrt{2}\right)}$. By the difference of squares, the denominator reduces to $-4$ and the fraction is $\frac{2-2\sqrt{2}}{-4}=\frac{\sqrt{2}-1}{2}$ which is $\boxed{\textbf{(A) } \frac{\sqrt{2}-1}{2}}$.

~Explanation by JH. L

Solution 2

The area of a regular octagon is $2(1+\sqrt{2})a^2$ where $a$ is the side. Hence the answer is obvious now.

See Also

2011 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 15 		Followed by
Problem 17
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 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2011_AMC_10B_Problems/Problem_16&oldid=226248"