document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210490>Question 1210490</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The centroid of triangle ABC is G.  Find BG. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852900 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852900\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The length of $BG$ is $\mathbf{\frac{2\sqrt{3}}{3}}$.\r<BR>\n" );
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document.write( "Here is the step-by-step calculation using the properties of an equilateral triangle and its centroid.\r<BR>\n" );
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document.write( "## 1. Find the Length of the Median ($\overline{BD}$)\r<BR>\n" );
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document.write( "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.\r<BR>\n" );
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document.write( "The altitude ($h$) of an equilateral triangle with side length $a$ is given by the formula:<BR>\n" );
document.write( "$$h = \frac{a\sqrt{3}}{2}$$\r<BR>\n" );
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document.write( "Substitute the side length $a=2$:<BR>\n" );
document.write( "$$BD = \frac{2\sqrt{3}}{2}$$<BR>\n" );
document.write( "$$BD = \mathbf{\sqrt{3}}$$\r<BR>\n" );
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document.write( "## 2. Find the Length of $\overline{BG}$\r<BR>\n" );
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document.write( "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 ($\overline{BD}$). \r<BR>\n" );
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document.write( "$$BG = \frac{2}{3} \cdot BD$$\r<BR>\n" );
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document.write( "Substitute the length of the median $BD = \sqrt{3}$:<BR>\n" );
document.write( "$$BG = \frac{2}{3} \cdot \sqrt{3}$$<BR>\n" );
document.write( "$$BG = \mathbf{\frac{2\sqrt{3}}{3}}$$\r<BR>\n" );
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document.write( "The length of $BG$ is $\frac{2\sqrt{3}}{3}$. </SPAN>\n" );
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