Question

the medians of \\(\triangle pqr\\) are \\(\overline{pt}\\), \\(\overline{qu}\\), and \\(\overline{rs}\\). they meet at a single point \\(v\\).
(in other words, \\(v\\) is the centroid of \\(\triangle pqr\\).)

suppose \\(qu = 21\\), \\(pv = 18\\), and \\(vs = 7\\).
find the following lengths.
note that the figure is not drawn to scale.

\\(rs = \\)
\\(qv = \\)
\\(vt = \\)

Explanation:

<pre_analysis>
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"concepts_used": [
"Centroid Theorem",
"Triangle Medians"
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"current_concepts": [
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<reasoning>

Calculate RS using the Centroid Theorem

\[

$$\begin{aligned} &RV = 2 \cdot VS \\ &VS = 7 \\ &RV = 2 \cdot 7 = 14 \\ &RS = RV + VS = 14 + 7 = 21 \end{aligned}$$

\]

Calculate QV using the Centroid Theorem

\[

$$\begin{aligned} &QV = \frac{2}{3} \cdot QU \\ &QU = 21 \\ &QV = \frac{2}{3} \cdot 21 = 14 \end{aligned}$$

\]

Calculate VT using the Centroid Theorem

\[

$$\begin{aligned} &PV = 2 \cdot VT \\ &PV = 18 \\ &VT = \frac{1}{2} \cdot PV = \frac{1}{2} \cdot 18 = 9 \end{aligned}$$

\]
</reasoning>

<answer>
The medians of \(\triangle PQR\) are \(\overline{PT}\), \(\overline{QU}\), and \(\overline{RS}\). They meet at a single point \(V\).
Suppose \(QU = 21\), \(PV = 18\), and \(VS = 7\).
Find the following lengths.

\(RS =\) <blank>21</blank>
\(QV =\) <blank>14</blank>
\(VT =\) <blank>9</blank>
</answer>

<post_analysis>
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"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Centroid Theorem"
]
}
</post_analysis>

Answer:

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"Triangle Medians"
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</pre_analysis>

<reasoning>

Calculate RS using the Centroid Theorem

\[

$$\begin{aligned} &RV = 2 \cdot VS \\ &VS = 7 \\ &RV = 2 \cdot 7 = 14 \\ &RS = RV + VS = 14 + 7 = 21 \end{aligned}$$

\]

Calculate QV using the Centroid Theorem

\[

$$\begin{aligned} &QV = \frac{2}{3} \cdot QU \\ &QU = 21 \\ &QV = \frac{2}{3} \cdot 21 = 14 \end{aligned}$$

\]

Calculate VT using the Centroid Theorem

\[

$$\begin{aligned} &PV = 2 \cdot VT \\ &PV = 18 \\ &VT = \frac{1}{2} \cdot PV = \frac{1}{2} \cdot 18 = 9 \end{aligned}$$

\]
</reasoning>

<answer>
The medians of \(\triangle PQR\) are \(\overline{PT}\), \(\overline{QU}\), and \(\overline{RS}\). They meet at a single point \(V\).
Suppose \(QU = 21\), \(PV = 18\), and \(VS = 7\).
Find the following lengths.

\(RS =\) <blank>21</blank>
\(QV =\) <blank>14</blank>
\(VT =\) <blank>9</blank>
</answer>

<post_analysis>
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"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Centroid Theorem"
]
}
</post_analysis>