You can put this solution on YOUR website! The length of $BG$ is $\mathbf{\frac{2\sqrt{3}}{3}}$.
Here is the step-by-step calculation using the properties of an equilateral triangle and its centroid.
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## 1. Find the Length of the Median ($\overline{BD}$)
In an **equilateral triangle** ($\triangle ABC$ with side length $a=2$), the median drawn from any vertex (like $B$) is also the altitude. Let $D$ be the midpoint of side $\overline{AC}$. $\overline{BD}$ is the median/altitude.
The altitude ($h$) of an equilateral triangle with side length $a$ is given by the formula:
$$h = \frac{a\sqrt{3}}{2}$$
Substitute the side length $a=2$:
$$BD = \frac{2\sqrt{3}}{2}$$
$$BD = \mathbf{\sqrt{3}}$$
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## 2. Find the Length of $\overline{BG}$
The centroid ($G$) divides any median into two segments with a ratio of **2:1**. The segment connecting the vertex ($B$) to the centroid ($G$) is two-thirds ($\frac{2}{3}$) the length of the entire median ($\overline{BD}$).
[Image of a triangle and its medial triangle with area ratio labeled]
$$BG = \frac{2}{3} \cdot BD$$
Substitute the length of the median $BD = \sqrt{3}$:
$$BG = \frac{2}{3} \cdot \sqrt{3}$$
$$BG = \mathbf{\frac{2\sqrt{3}}{3}}$$
The length of $BG$ is $\frac{2\sqrt{3}}{3}$.
The problem's formulation is incomplete and can not be considered as a true/proper Math problem.
Is not a subject for discussion or solution.
Is the subject to throw to a garbage bin.
The fact that @CPhill is using artificial intelligence to solve this problem is the attempt
to deceive a reader and tells about his irresponsible attitude towards his duties.
(