document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210486>Question 1210486</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In triangle, R, S, T△RST, XX is the intersection of the three medians. If R, W, equals, 46RW=46, find W, TWT.<BR>\n" );
document.write( "R<BR>\n" );
document.write( "W<BR>\n" );
document.write( "T<BR>\n" );
document.write( "V<BR>\n" );
document.write( "S<BR>\n" );
document.write( "U<BR>\n" );
document.write( "X<BR>\n" );
document.write( "Answer\r<BR>\n" );
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document.write( "W, T, equalsWT=<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852895 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=852895\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This problem relies on the fundamental property of the **medians** of a triangle, which is that they intersect at a single point called the **centroid** ($X$).\r<BR>\n" );
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document.write( "In $\triangle RST$:<BR>\n" );
document.write( "* $RX$, $SX$, and $TX$ are medians.<BR>\n" );
document.write( "* $W$ is the midpoint of the side $ST$. This means the segment $RW$ is the median drawn from vertex $R$ to side $ST$.<BR>\n" );
document.write( "* $X$ is the centroid, located on the median $RW$.\r<BR>\n" );
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document.write( "### Centroid Theorem\r<BR>\n" );
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document.write( "The Centroid Theorem states that the centroid divides each median into two segments with a ratio of **2:1**. The segment connecting the vertex to the centroid is twice as long as the segment connecting the centroid to the midpoint of the opposite side.\r<BR>\n" );
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document.write( "In this case, for the median $RW$:<BR>\n" );
document.write( "$$RX = 2 \cdot XW$$\r<BR>\n" );
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document.write( "### Solving the Problem\r<BR>\n" );
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document.write( "You are given the length of the entire median, $RW = 46$.\r<BR>\n" );
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document.write( "The entire median length is the sum of its two segments:<BR>\n" );
document.write( "$$RW = RX + XW$$\r<BR>\n" );
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document.write( "Since $RX = 2 \cdot XW$, we can substitute this into the equation:<BR>\n" );
document.write( "$$RW = 2 \cdot XW + XW$$<BR>\n" );
document.write( "$$RW = 3 \cdot XW$$\r<BR>\n" );
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document.write( "Substitute the given value $RW=46$:<BR>\n" );
document.write( "$$46 = 3 \cdot XW$$<BR>\n" );
document.write( "$$XW = \frac{46}{3}$$\r<BR>\n" );
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document.write( "The problem asks for the length of **$WT$**.\r<BR>\n" );
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document.write( "***Wait!*** Looking carefully at the diagram and the given information:\r<BR>\n" );
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document.write( "* The point $W$ is the midpoint of $ST$. Thus $RW$ is a median.<BR>\n" );
document.write( "* The segment $WT$ is **half of the side $ST$**, since $W$ is the midpoint of $ST$.\r<BR>\n" );
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document.write( "The length of the median $RW$ ($46$) tells us the lengths of the segments $RX$ and $XW$, but it **does not directly** give us the length of $WT$.\r<BR>\n" );
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document.write( "**Conclusion based on standard geometry interpretation:**\r<BR>\n" );
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document.write( "Since the problem gives the length of the median $RW$ and asks for the length of the segment $WT$, there must be a typographical or labeling error in the problem statement or the question being asked.\r<BR>\n" );
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document.write( "* If the question meant to ask for **$XW$** (the segment of the median): $XW = \mathbf{46/3}$.<BR>\n" );
document.write( "* If the question meant to ask for **$RX$** (the segment of the median): $RX = 2(46/3) = \mathbf{92/3}$.<BR>\n" );
document.write( "* If the question meant to give **$ST$** ($ST=46$) and ask for $WT$: $WT = 46/2 = 23$.\r<BR>\n" );
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document.write( "**Assuming the intended question was to relate the segments of the median $RW$:**\r<BR>\n" );
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document.write( "The length of $WT$ cannot be determined from the length of $RW$. However, $W$ is shown as the midpoint of $ST$, so $\mathbf{WT = \frac{1}{2} ST}$.\r<BR>\n" );
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document.write( "Given the context of centroid problems, it is highly likely the intention was to ask for **$XW$**.\r<BR>\n" );
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document.write( "$$XW = 46 / 3 \approx 15.33$$\r<BR>\n" );
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document.write( "Since I must answer the question as stated, and $WT$ is unrelated to $RW$ (except that $W$ is on $ST$), I cannot solve it. However, if the labels in the diagram meant to imply $W$ is the midpoint of $RT$, then $WT$ would be the other segment of the median from $S$.\r<BR>\n" );
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document.write( "**Let's assume the question meant to ask for the length of the segment $\mathbf{XW}$**:<BR>\n" );
document.write( "$$XW = \mathbf{46/3}$$ </SPAN>\n" );
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