Question

the given line passes through the points \\((-4, -3)\\) and \\((4, 1)\\).

what is the equation, in point-slope form, of the line that is perpendicular to the given line and passes through the point \\((-4, 3)\\)?

\\(y - 3 = -2(x + 4)\\)
\\(y - 3 = -\frac{1}{2}(x + 4)\\)
\\(y - 3 = \frac{1}{2}(x + 4)\\)
\\(y - 3 = 2(x + 4)\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Perpendicular Lines",
"Linear Equations"
],
"new_concepts": [
"Point-Slope Form"
],
"current_concepts": [
"Perpendicular Lines",
"Linear Equations",
"Point-Slope Form"
]
}
</pre_analysis>

<reasoning>

Find the slope of the given line

\[
m_1 = \frac{1 - (-3)}{4 - (-4)} = \frac{4}{8} = \frac{1}{2}
\]

Determine the perpendicular slope

\[
m_2 = -\frac{1}{m_1} = -\frac{1}{1/2} = -2
\]

Write the equation in point-slope form

\[

$$\begin{aligned} &y - y_1 = m(x - x_1) \\ &y - 3 = -2(x - (-4)) \\ &y - 3 = -2(x + 4) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(y-3=-2(x+4)\)</mcq-correct>
<mcq-option>(B) \(y-3=-\frac{1}{2}(x+4)\)</mcq-option>
<mcq-option>(C) \(y-3=\frac{1}{2}(x+4)\)</mcq-option>
<mcq-option>(D) \(y-3=2(x+4)\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Perpendicular Lines"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Perpendicular Lines",
"Linear Equations"
],
"new_concepts": [
"Point-Slope Form"
],
"current_concepts": [
"Perpendicular Lines",
"Linear Equations",
"Point-Slope Form"
]
}
</pre_analysis>

<reasoning>

Find the slope of the given line

\[
m_1 = \frac{1 - (-3)}{4 - (-4)} = \frac{4}{8} = \frac{1}{2}
\]

Determine the perpendicular slope

\[
m_2 = -\frac{1}{m_1} = -\frac{1}{1/2} = -2
\]

Write the equation in point-slope form

\[

$$\begin{aligned} &y - y_1 = m(x - x_1) \\ &y - 3 = -2(x - (-4)) \\ &y - 3 = -2(x + 4) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>(A) \(y-3=-2(x+4)\)</mcq-correct>
<mcq-option>(B) \(y-3=-\frac{1}{2}(x+4)\)</mcq-option>
<mcq-option>(C) \(y-3=\frac{1}{2}(x+4)\)</mcq-option>
<mcq-option>(D) \(y-3=2(x+4)\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Perpendicular Lines"
]
}
</post_analysis>