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| | A square and an equilateral triangle have the same perimeter. Let <math>A</math> be the area of the circle circumscribed about the square and <math>B</math> the area of the circle circumscribed around the triangle. Find <math>A/B</math>. | | A square and an equilateral triangle have the same perimeter. Let <math>A</math> be the area of the circle circumscribed about the square and <math>B</math> the area of the circle circumscribed around the triangle. Find <math>A/B</math>. |
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| − | <math> \mathrm{(A) \ } \frac{9}{16}\qquad \mathrm{(B) \ } \frac{3}{4}\qquad \mathrm{(C) \ } \frac{27}{32}\qquad \mathrm{(D) \ } \frac{3\sqrt{6}}{8}\qquad \mathrm{(E) \ } 1 </math>
| + | $ \mathrm{(A) \ |
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| − | == Solution ==
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| − | Suppose that the common perimeter is <math>P</math>.
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| − | Then, the side lengths of the square and triangle, respectively, are <math>\frac{P}{4}</math> and <math>\frac{P}{3}</math>
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| − | The circle circumscribed about the square has a diameter equal to the diagonal of the square, which is <math>\frac{P\sqrt{2}}{4}</math>
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| − | Therefore, the radius is <math>\frac{P\sqrt{2}}{8}</math> and the area of the circle is
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| − | <math>\pi \cdot \left(\frac{P\sqrt{2}}{8}\right)^2 = \pi \cdot \frac{2P^2}{64}=\boxed{\frac{P^2 \pi}{32}=A}</math>
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| − | Now consider the circle circumscribed around the equilateral triangle. Due to symmetry, the circle must share a center with the equilateral triangle. The radius of the circle is simply the distance from the center of the triangle to a vertex.
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| − | This distance is <math>\frac{2}{3}</math> of an altitude. By <math>30-60-90</math> right triangle properties, the altitude is <math>\frac{\sqrt{3}}{2} \cdot s</math> where s is the side.
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| − | So, the radius is <math>\frac{2}{3} \cdot \frac{\sqrt{3}}{2} \cdot \frac{P}{3} = \frac{P\sqrt{3}}{9}</math>
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| − | The area of the circle is <math>\pi \cdot \left(\frac{P\sqrt{3}}{9}\right)^2=\pi \cdot \frac{3P^2}{81}=\boxed{\frac{P^2\pi}{27}=B}</math>
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| − | So, <math>\frac{A}{B}=\frac{\frac{P^2 \pi}{32}}{\frac{P^2 \pi}{27}}=\frac{P^2 \pi}{32} \cdot \frac{27}{P^2\pi}=\boxed{\frac{27}{32} \implies \mathrm{(C) \ } \frac{27}{32}}</math>
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| − | == See Also ==
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| − | {{AMC12 box|year=2003|ab=A|num-b=10|num-a=12}}
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| − | {{MAA Notice}}
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be the area of the circle circumscribed about the square and 
the area of the circle circumscribed around the triangle. Find 
.
