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Let $DEF$ be an equilateral triangle with side length $3.$ At random, a point $G$ is chosen inside the triangle. Compute the probability that the length $DG$ is less than or equal to $1.$

 Jul 27, 2021
 #1
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The probability is pi/12.

 Jul 27, 2021
 #2
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The valid points $G$ are inside the circular sector centered at $D$. This sector has radius $1$ and angle $60$. Hence it's area is $\frac{\pi}{6}$. The area of the entire triangle is $\frac{9\sqrt{3}}{4}$, so by the principle of geometric probability, the answer is $\frac{4\pi}{54\sqrt{3}}=\frac{2\sqrt{3}\pi}{81}$

 Jul 27, 2021

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