SOLUTION: In triangle, R, S, T△RST, XX is the intersection of the three medians. If R, W, equals, 46RW=46, find W, TWT. R W T V S U X Answer W, T, equalsWT=

Question 1210486: In triangle, R, S, T△RST, XX is the intersection of the three medians. If R, W, equals, 46RW=46, find W, TWT.
R
W
T
V
S
U
X
Answer
W, T, equalsWT=

Answer by CPhill(2138) (Show Source): You can put this solution on YOUR website!
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** ($X$).
In $\triangle RST$:
* $RX$, $SX$, and $TX$ are medians.
* $W$ is the midpoint of the side $ST$. This means the segment $RW$ is the median drawn from vertex $R$ to side $ST$.
* $X$ is the centroid, located on the median $RW$.
### Centroid Theorem
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.
In this case, for the median $RW$:
$$RX = 2 \cdot XW$$
### Solving the Problem
You are given the length of the entire median, $RW = 46$.
The entire median length is the sum of its two segments:
$$RW = RX + XW$$
Since $RX = 2 \cdot XW$, we can substitute this into the equation:
$$RW = 2 \cdot XW + XW$$
$$RW = 3 \cdot XW$$
Substitute the given value $RW=46$:
$$46 = 3 \cdot XW$$
$$XW = \frac{46}{3}$$
The problem asks for the length of **$WT$**.
***Wait!*** Looking carefully at the diagram and the given information:
* The point $W$ is the midpoint of $ST$. Thus $RW$ is a median.
* The segment $WT$ is **half of the side $ST$**, since $W$ is the midpoint of $ST$.
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 $WT$.
**Conclusion based on standard geometry interpretation:**
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.
* If the question meant to ask for **$XW$** (the segment of the median): $XW = \mathbf{46/3}$.
* If the question meant to ask for **$RX$** (the segment of the median): $RX = 2(46/3) = \mathbf{92/3}$.
* If the question meant to give **$ST$** ($ST=46$) and ask for $WT$: $WT = 46/2 = 23$.
**Assuming the intended question was to relate the segments of the median $RW$:**
The length of $WT$ cannot be determined from the length of $RW$. However, $W$ is shown as the midpoint of $ST$, so $\mathbf{WT = \frac{1}{2} ST}$.
Given the context of centroid problems, it is highly likely the intention was to ask for **$XW$**.
$$XW = 46 / 3 \approx 15.33$$
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 $S$.
**Let's assume the question meant to ask for the length of the segment $\mathbf{XW}$**:
$$XW = \mathbf{46/3}$$
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