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Difference between revisions of "2009 AMC 10A Problems/Problem 21"

(→‎Solution)
(fixed a bug in the solution)
Line 43: Line 43:
 
draw( (0,sqrt(2)) -- (0,1+sqrt(2)) );
 
draw( (0,sqrt(2)) -- (0,1+sqrt(2)) );
 
draw( (0,-sqrt(2)) -- (0,-1-sqrt(2)) );
 
draw( (0,-sqrt(2)) -- (0,-1-sqrt(2)) );
 +
draw( (0,sqrt(2)) -- (0,-sqrt(2)), dashed );
 
</asy>
 
</asy>
  
−
It is now obvious that the radius <math>R</math> of the large circle can be expressed using the radius <math>r</math> of the small circles as <math>R=2r + 2r\sqrt 2</math>. Then the area of the large circle is <math>L = \pi R^2 = \pi (2r)^2 (1+\sqrt 2)^2 = 4\pi r^2 (3+2\sqrt 2)</math>.
+
Out of symmetry, the quadrilateral in the center must be a square. Its side is obviously <math>2r</math>, and therefore its diagonal is <math>2r\sqrt{2}</math>. We can now compute the length of the vertical diameter of the large circle as <math>2r + 2r\sqrt{2}</math>. Hence <math>2R=2r + 2r\sqrt{2}</math>, and thus <math>R=r+r\sqrt{2}=r(1+\sqrt{2})</math>.
 +
 
 +
Then the area of the large circle is <math>L = \pi R^2 = \pi r^2 (1+\sqrt 2)^2 = \pi r^2 (3+2\sqrt 2)</math>.
 
The area of four small circles is <math>S = 4\pi r^2</math>. Hence their ratio is:
 
The area of four small circles is <math>S = 4\pi r^2</math>. Hence their ratio is:
  
Line 51: Line 54:
 
\begin{align*}
 
\begin{align*}
 
\frac SL  
 
\frac SL  
−
& = \frac{4\pi r^2}{4\pi r^2 (3+2\sqrt 2)} \\
+
& = \frac{4\pi r^2}{\pi r^2 (3+2\sqrt 2)} \\
−
& = \frac 1{3+2\sqrt 2} \\
+
& = \frac 4{3+2\sqrt 2} \\
−
& = \frac 1{3+2\sqrt 2} \cdot \frac{3-2\sqrt 2}{3 - 2\sqrt 2} \\
+
& = \frac 4{3+2\sqrt 2} \cdot \frac{3-2\sqrt 2}{3 - 2\sqrt 2} \\
−
& = \frac{3 - 2\sqrt 2}{3^2 - (2\sqrt 2)^2} \\
+
& = \frac{4(3 - 2\sqrt 2)}{3^2 - (2\sqrt 2)^2} \\
−
& = \frac{3 - 2\sqrt 2}1 \\
+
& = \frac{4(3 - 2\sqrt 2)}1 \\
−
& = \boxed{3 - 2\sqrt 2}
+
& = \boxed{4(3 - 2\sqrt 2)}
−
 
 
−
 
 
−
But the answer is C.
 
 
\end{align*}
 
\end{align*}
 
</cmath>
 
</cmath>

Revision as of 12:57, 25 February 2009

Problem

Many Gothic cathedrals have windows with portions containing a ring of congruent circles that are circumscribed by a larger circle, In the figure shown, the number of smaller circles is four. What is the ratio of the sum of the areas of the four smaller circles to the area of the larger circle?

[asy] unitsize(6mm); defaultpen(linewidth(.8pt)); draw(Circle((0,0),1+sqrt(2))); draw(Circle((sqrt(2),0),1)); draw(Circle((0,sqrt(2)),1)); draw(Circle((-sqrt(2),0),1)); draw(Circle((0,-sqrt(2)),1)); [/asy]

$\mathrm{(A)}\ 3-2\sqrt2 \qquad \mathrm{(B)}\ 2-\sqrt2 \qquad \mathrm{(C)}\ 4(3-2\sqrt2) \qquad \mathrm{(D)}\ \frac12(3-\sqrt2) \qquad \mathrm{(E)}\ 2\sqrt2-2$


Solution

Draw some of the radii of the small circles as in the picture below.

[asy] unitsize(12mm); defaultpen(linewidth(.8pt)); draw(Circle((0,0),1+sqrt(2))); draw(Circle((sqrt(2),0),1)); draw(Circle((0,sqrt(2)),1)); draw(Circle((-sqrt(2),0),1)); draw(Circle((0,-sqrt(2)),1)); draw( (sqrt(2),0) -- (0,sqrt(2)) -- (-sqrt(2),0) -- (0,-sqrt(2)) -- cycle ); draw( (0,sqrt(2)) -- (0,1+sqrt(2)) ); draw( (0,-sqrt(2)) -- (0,-1-sqrt(2)) ); draw( (0,sqrt(2)) -- (0,-sqrt(2)), dashed ); [/asy]

Out of symmetry, the quadrilateral in the center must be a square. Its side is obviously $2r$, and therefore its diagonal is $2r\sqrt{2}$. We can now compute the length of the vertical diameter of the large circle as $2r + 2r\sqrt{2}$. Hence $2R=2r + 2r\sqrt{2}$, and thus $R=r+r\sqrt{2}=r(1+\sqrt{2})$.

Then the area of the large circle is $L = \pi R^2 = \pi r^2 (1+\sqrt 2)^2 = \pi r^2 (3+2\sqrt 2)$. The area of four small circles is $S = 4\pi r^2$. Hence their ratio is:

\begin{align*} \frac SL & = \frac{4\pi r^2}{\pi r^2 (3+2\sqrt 2)} \\ & = \frac 4{3+2\sqrt 2} \\ & = \frac 4{3+2\sqrt 2} \cdot \frac{3-2\sqrt 2}{3 - 2\sqrt 2} \\ & = \frac{4(3 - 2\sqrt 2)}{3^2 - (2\sqrt 2)^2} \\ & = \frac{4(3 - 2\sqrt 2)}1 \\ & = \boxed{4(3 - 2\sqrt 2)} \end{align*}

See Also

2009 AMC 10A (Problems • Answer Key • Resources)
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All AMC 10 Problems and Solutions
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