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| <math>\textbf{(A) }3\sqrt{3}-\pi\qquad\textbf{(B) }4\sqrt{3}-\frac{4\pi}{3}\qquad\textbf{(C) }2\sqrt{3}\qquad\textbf{(D) }4\sqrt{3}-\frac{2\pi}{3}\qquad\textbf{(E) }4+\frac{4\pi}{3}</math> | | <math>\textbf{(A) }3\sqrt{3}-\pi\qquad\textbf{(B) }4\sqrt{3}-\frac{4\pi}{3}\qquad\textbf{(C) }2\sqrt{3}\qquad\textbf{(D) }4\sqrt{3}-\frac{2\pi}{3}\qquad\textbf{(E) }4+\frac{4\pi}{3}</math> |
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| ==Solution==
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| Let the centers of the circles containing arcs <math>\overarc{SR}</math> and <math>\overarc{TR}</math> be <math>X</math> and <math>Y</math>, respectively. Extend <math>\overline{US}</math> and <math>\overline{UT}</math> to <math>X</math> and <math>Y</math>, and connect point <math>X</math> with point <math>Y</math>.
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| <asy>
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| unitsize(1 cm);
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| pair U,S,T,R,X,Y;
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| U =(2,3.464);
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| S=(1,1.732);
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| T=(3,1.732);
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| R=(2,0);
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| X=(0,0);
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| Y=(4,0);
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| draw(U--S);
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| draw(S--U--T);
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| draw(S--X--Y--T,red);
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| draw(arc(X,R,S),red);
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| draw(arc(Y,T,R),red);
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| label("$U$",U, N);
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| label("$S$", S, W);
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| label("$T$", T, E);
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| label("$R$", R, S);
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| label("$X$",X, W);
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| label("$Y$", Y, E);
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| </asy>
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| We can clearly see that <math>\triangle UXY</math> is an equilateral triangle, because the problem states that <math>m\angle TUS = 60^\circ</math>. We can figure out that <math>m\angle SXR= 60^\circ</math> and <math>m\angle TYR = 60^\circ</math> because they are <math>\frac{1}{6}</math> of a circle. The area of the figure is equal to <math>[\triangle UXY]</math> minus the combined area of the <math>2</math> sectors of the circles(in red). Using the area formula for an equilateral triangle, <math>\frac{a^2\sqrt{3}}{4},</math> where <math>a</math> is the side length of the equilateral triangle, <math>[\triangle UXY]</math> is <math>\frac{\sqrt 3}{4} \cdot 4^2 = 4\sqrt 3.</math> The combined area of the <math>2</math> sectors is <math>2\cdot\frac16\cdot\pi r^2</math>, which is <math>\frac 13\pi \cdot 2^2 = \frac{4\pi}{3}.</math> Thus, our final answer is <math>\boxed{\textbf{(B)}\ 4\sqrt{3}-\frac{4\pi}{3}}.</math>
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| ==See Also== | | ==See Also== |
Revision as of 22:23, 9 November 2019
Problem 25
In the figure shown, 
and 
are line segments each of length 2, and 
. Arcs 
and 
are each one-sixth of a circle with radius 2. What is the area of the region shown?
![[asy]draw((1,1.732)--(2,3.464)--(3,1.732)); draw(arc((0,0),(2,0),(1,1.732))); draw(arc((4,0),(3,1.732),(2,0))); label("$U$", (2,3.464), N); label("$S$", (1,1.732), W); label("$T$", (3,1.732), E); label("$R$", (2,0), S);[/asy]](//latex.artofproblemsolving.com/7/9/f/79f3ffcfa12364f64959c7aeadc809ba8d2aa6ac.png)
See Also
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: Unable to save thumbnail to destination
and 
are line segments each of length 2, and 
. Arcs 
and 
are each one-sixth of a circle with radius 2. What is the area of the region shown?
![[asy]draw((1,1.732)--(2,3.464)--(3,1.732)); draw(arc((0,0),(2,0),(1,1.732))); draw(arc((4,0),(3,1.732),(2,0))); label("$U$", (2,3.464), N); label("$S$", (1,1.732), W); label("$T$", (3,1.732), E); label("$R$", (2,0), S);[/asy]](http://latex.artofproblemsolving.com/7/9/f/79f3ffcfa12364f64959c7aeadc809ba8d2aa6ac.png)
