Problem

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 1

label(" $Y$ ", Y, E); </asy> We can clearly see that $\triangle UXY$ is an equilateral triangle, because the problem states that $m\angle TUS = 60^\circ$ . We can figure out that $m\angle SXR= 60^\circ$ and $m\angle TYR = 60^\circ$ because they are $\frac{1}{6}$ of a circle. The area of the figure is equal to $[\triangle UXY]$ minus the combined area of the $2$ sectors of the circles (in red). Using the area formula for an equilateral triangle, $\frac{a^2\sqrt{3}}{4},$ where $a$ is the side length of the equilateral triangle, $[\triangle UXY]$ is $\frac{\sqrt 3}{4} \cdot 4^2 = 4\sqrt 3.$ The combined area of the $2$ sectors is $2\cdot\frac16\cdot\pi r^2$ , which is $\frac 13\pi \cdot 2^2 = \frac{4\pi}{3}.$ Thus, our final answer is $\boxed{\textbf{(B)}\ 4\sqrt{3}-\frac{4\pi}{3}}.$

Video Solutions

https://youtu.be/LT4gyH--328

https://youtu.be/wc5rGulTTR8

- Happytwin

https://youtu.be/aE0oAq4Q_Ks

https://euclideanmathcircle.wixsite.com/emc1/videos?wix-vod-video-id=3a7970c3cd01453aa4263a8be7998588&wix-vod-comp-id=comp-kn8844mv

https://youtu.be/sVclz6EmpEU

~savannahsolver

Solution 3 & Video

First, we must make it like an equilateral triangle. If you add two of the same arcs, you will get an equilateral triangle. The goal is to find ONE of the arcs and multiply it by two. Then subtract it from the equilateral triangle. If you look at the triangle, T and S are midpoints of the equilateral triangle. So now we can find the arc.These arcs are $\frac{2}{6}\pi$ so now you multiply by 2 so $\frac{4}{3}\pi$ . And subtract. Equilateral's area is $4\sqrt{3}$ . So now you do $4 \sqrt 3 - \frac{4}{3} \pi$ . That is the answer and thank you.

~ math.is.hard.to.understand

Here's a video to understand it (credits to ~ pi_is_3.14)

https://youtu.be/j3QSD5eDpzU?t=1350

2017 AMC 8 (Problems • Answer Key • Resources)
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2017 AMC 8 Problems/Problem 25

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