Question

what is the equation, in point-slope form, of the line that is perpendicular to the given line and passes through the point \\((2, 5)\\)?

\\(y + 5 = x + 2\\)
\\(y - 2 = x - 5\\)
\\(y - 5 = -(x - 2)\\)
\\(y + 2 = -(x + 5)\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": true,
"concepts_used": [
"Slope of a Line",
"Point-Slope Form"
],
"new_concepts": [
"Perpendicular Line Equations"
],
"current_concepts": [
"Slope of a Line",
"Point-Slope Form",
"Perpendicular Line Equations"
]
}
</pre_analysis>

<reasoning>

Find the slope of the given line

Using the Slope of a Line knowledge point
\[
m_1 = \frac{2 - (-1)}{4 - 1} = \frac{3}{3} = 1
\]

Determine the perpendicular slope

The slope \(m_2\) of a line perpendicular to a line with slope \(m_1\) satisfies:
\[
m_2 = -\frac{1}{m_1} = -\frac{1}{1} = -1
\]

Write the equation in point-slope form

Using the Point-Slope Form knowledge point
\[

$$\begin{aligned} &y - y_1 = m_2(x - x_1)\\ &y - 5 = -1(x - 2)\\ &y - 5 = -(x - 2) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(y + 5 = x + 2\)</mcq-option>
<mcq-option>(B) \(y - 2 = x - 5\)</mcq-option>
<mcq-correct>(C) \(y - 5 = -(x - 2)\)</mcq-correct>
<mcq-option>(D) \(y + 2 = -(x + 5)\)</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "line",
"params": [[1, -1], [4, 2]],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 2,
"name": "Given Line",
"withLabel": false
}
},
{
"type": "point",
"params": [[1, -1]],
"properties": {
"name": "(1, -1)",
"size": 3,
"color": "#5583F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[4, 2]],
"properties": {
"name": "(4, 2)",
"size": 3,
"color": "#5583F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[2, 5]],
"properties": {
"name": "(2, 5)",
"size": 4,
"color": "#F2557F",
"withLabel": true
}
},
{
"type": "line",
"params": [[2, 5], [5, 2]],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"strokeDash": 2,
"name": "Perpendicular Line",
"withLabel": false
}
}
]
}
</plot>

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

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": true,
"concepts_used": [
"Slope of a Line",
"Point-Slope Form"
],
"new_concepts": [
"Perpendicular Line Equations"
],
"current_concepts": [
"Slope of a Line",
"Point-Slope Form",
"Perpendicular Line Equations"
]
}
</pre_analysis>

<reasoning>

Find the slope of the given line

Using the Slope of a Line knowledge point
\[
m_1 = \frac{2 - (-1)}{4 - 1} = \frac{3}{3} = 1
\]

Determine the perpendicular slope

The slope \(m_2\) of a line perpendicular to a line with slope \(m_1\) satisfies:
\[
m_2 = -\frac{1}{m_1} = -\frac{1}{1} = -1
\]

Write the equation in point-slope form

Using the Point-Slope Form knowledge point
\[

$$\begin{aligned} &y - y_1 = m_2(x - x_1)\\ &y - 5 = -1(x - 2)\\ &y - 5 = -(x - 2) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(y + 5 = x + 2\)</mcq-option>
<mcq-option>(B) \(y - 2 = x - 5\)</mcq-option>
<mcq-correct>(C) \(y - 5 = -(x - 2)\)</mcq-correct>
<mcq-option>(D) \(y + 2 = -(x + 5)\)</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "line",
"params": [[1, -1], [4, 2]],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 2,
"name": "Given Line",
"withLabel": false
}
},
{
"type": "point",
"params": [[1, -1]],
"properties": {
"name": "(1, -1)",
"size": 3,
"color": "#5583F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[4, 2]],
"properties": {
"name": "(4, 2)",
"size": 3,
"color": "#5583F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[2, 5]],
"properties": {
"name": "(2, 5)",
"size": 4,
"color": "#F2557F",
"withLabel": true
}
},
{
"type": "line",
"params": [[2, 5], [5, 2]],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"strokeDash": 2,
"name": "Perpendicular Line",
"withLabel": false
}
}
]
}
</plot>

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