text: 2017 AMC 8 Problems/Problem 25
  
  Problem 25
  
  In the figure shown, $\overline{US}$ and $\overline{UT}$ are line segments each of length 2, and $m\angle TUS = 60^\circ$. Arcs $\overarc{TR}$ and $\overarc{SR}$ are each one-sixth of a circle with radius 2. What is the area of the region shown?
  
  $[asy]draw((1,1.732)--(2,3.464)--(3,1.732)); draw(arc((0,0),(2,0),(1,1.732))); draw(arc((4,0),(3,1.732),(2,0))); label("U", (2,3.464), N); label("S", (1,1.732), W); label("T", (3,1.732), E); label("R", (2,0), S);[/asy]$
  
  $\textbf{(A) }3\sqrt{3}-\pi\qquad\textbf{(B) }4\sqrt{3}-\frac{4\pi}{3}\qquad\textbf{(C) }2\sqrt{3}\qquad\textbf{(D) }4\sqrt{3}-\frac{2\pi}{3}\qquad\textbf{(E) }4+\frac{4\pi}{3}$
  
  Solution
  
  We can extend $\overline{US}$ and $\overline{UT}$ into an equilateral triangle of side length $4$ with area $\dfrac{\sqrt{3}}{4}\cdot4^2=4\sqrt{3}$. We then must subtract the two 1/6-circles, which each have radius $2$, so their total area is $2\cdot\dfrac{\pi\cdot2^2}{6}=\dfrac{4\pi}{3}$. Our answer is therefore $\boxed{\textbf{(B)}\ 4\sqrt{3}-\dfrac{4\pi}{3}}$.
explanation: The document contains a math problem from the 2017 AMC 8 competition, specifically Problem 25. The problem asks for the area of a region defined by line segments and arcs. The document also includes a detailed solution to the problem, leading to the final answer.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: 2017 AMC 8 Problems/Problem 25
  
  Problem 25
  
  In the figure shown, $\overline{US}$ and $\overline{UT}$ are line segments each of length 2, and $m\angle TUS = 60^\circ$. Arcs $\overarc{TR}$ and $\overarc{SR}$ are each one-sixth of a circle with radius 2. What is the area of the region shown?
  
  $[asy]draw((1,1.732)--(2,3.464)--(3,1.732)); draw(arc((0,0),(2,0),(1,1.732))); draw(arc((4,0),(3,1.732),(2,0))); label("U", (2,3.464), N); label("S", (1,1.732), W); label("T", (3,1.732), E); label("R", (2,0), S);[/asy]$
  
  $\textbf{(A) }3\sqrt{3}-\pi\qquad\textbf{(B) }4\sqrt{3}-\frac{4\pi}{3}\qquad\textbf{(C) }2\sqrt{3}\qquad\textbf{(D) }4\sqrt{3}-\frac{2\pi}{3}\qquad\textbf{(E) }4+\frac{4\pi}{3}$
  
  Solution
  
  We can extend $\overline{US}$ and $\overline{UT}$ into an equilateral triangle of side length $4$ with area $\dfrac{\sqrt{3}}{4}\cdot4^2=4\sqrt{3}$. We then must subtract the two 1/6-circles, which each have radius $2$, so their total area is $2\cdot\dfrac{\pi\cdot2^2}{6}=\dfrac{4\pi}{3}$. Our answer is therefore $\boxed{\textbf{(B)}\ 4\sqrt{3}-\dfrac{4\pi}{3}}$.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math problem from the 2017 AMC 8 competition, specifically Problem 25. The problem asks for the area of a region defined by line segments and arcs. The document also includes a detailed solution to the problem, leading to the final answer.
  
  Label: question-and-answer