Question

what are the \\(x\\)- and \\(y\\)- coordinates of point p on the directed line segment from a to b such that p is \\(\frac{1}{3}\\) the length of the line segment from a to b?

\\x = \left(\frac{m}{m+n}\
ight)(x_2 - x_1) + x_1\\
\\y = \left(\frac{m}{m+n}\
ight)(y_2 - y_1) + y_1\\

(1, 5)
(0, 3)
(-4, -5)
(-5, -7)

Explanation:

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"Section Formula"
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"new_concepts": [],
"current_concepts": [
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"Midpoint Partition Ratio"
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</pre_analysis>

<reasoning>

Identify coordinates and partition ratio

\[

$$\begin{aligned} &A(x_1, y_1) = (-8, -13)\\ &B(x_2, y_2) = (4, 11)\\ &\frac{m}{m+n} = \frac{1}{3} \end{aligned}$$

\]

Calculate the x-coordinate of point P

\[

$$\begin{aligned} x &= \frac{1}{3}(4 - (-8)) + (-8)\\ x &= \frac{1}{3}(12) - 8\\ x &= 4 - 8 = -4 \end{aligned}$$

\]

Calculate the y-coordinate of point P

\[

$$\begin{aligned} y &= \frac{1}{3}(11 - (-13)) + (-13)\\ y &= \frac{1}{3}(24) - 13\\ y &= 8 - 13 = -5 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \((1, 5)\)</mcq-option>
<mcq-option>(B) \((0, 3)\)</mcq-option>
<mcq-correct>(C) \((-4, -5)\)</mcq-correct>
<mcq-option>(D) \((-5, -7)\)</mcq-option>
</answer>

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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Directed Line Segments"
]
}
</post_analysis>

Answer:

<pre_analysis>
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"skills_matched": [
"step_cot",
"multiple_choice_formatter",
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"new_concepts": [],
"current_concepts": [
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}
</pre_analysis>

<reasoning>

Identify coordinates and partition ratio

\[

$$\begin{aligned} &A(x_1, y_1) = (-8, -13)\\ &B(x_2, y_2) = (4, 11)\\ &\frac{m}{m+n} = \frac{1}{3} \end{aligned}$$

\]

Calculate the x-coordinate of point P

\[

$$\begin{aligned} x &= \frac{1}{3}(4 - (-8)) + (-8)\\ x &= \frac{1}{3}(12) - 8\\ x &= 4 - 8 = -4 \end{aligned}$$

\]

Calculate the y-coordinate of point P

\[

$$\begin{aligned} y &= \frac{1}{3}(11 - (-13)) + (-13)\\ y &= \frac{1}{3}(24) - 13\\ y &= 8 - 13 = -5 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \((1, 5)\)</mcq-option>
<mcq-option>(B) \((0, 3)\)</mcq-option>
<mcq-correct>(C) \((-4, -5)\)</mcq-correct>
<mcq-option>(D) \((-5, -7)\)</mcq-option>
</answer>

<post_analysis>
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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Directed Line Segments"
]
}
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