Question

what are the coordinates of \\(a\\) after reflecting across the line \\(x = 3\\)?

Explanation:

<pre_analysis>
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"question_count": 1,
"skills_matched": [
"step_cot",
"explain_then_answer",
"quick"
],
"needs_drawing": true,
"concepts_used": [
"Coordinate Transformations"
],
"new_concepts": [
"Reflection Across Vertical Lines"
],
"current_concepts": [
"Coordinate Transformations",
"Reflection Across Vertical Lines"
]
}
</pre_analysis>

<reasoning>

Identify coordinates of point A

\[
A = (3, 2)
\]

Apply reflection rule across vertical line

\[

$$\begin{aligned} x' &= 2h - x \\ x' &= 2(3) - 3 = 3 \\ y' &= y = 2 \end{aligned}$$

\]

State coordinates of reflected point

\[
A' = (3, 2)
\]
</reasoning>

<answer>
\((3, 2)\)
</answer>

<plot>
{
"elements": [
{
"type": "line",
"params": [[3, -3], [3, 7]],
"properties": {
"name": "x = 3",
"strokeColor": "#F2557F",
"strokeWidth": 2,
"dash": 2,
"withLabel": true
}
},
{
"type": "polygon",
"params": [[3, 2], [2, 4], [6, 3], [1, 0]],
"properties": {
"fillColor": "rgba(140, 85, 242, 0.15)",
"strokeColor": "#8C55F2",
"strokeWidth": 2
}
},
{
"type": "point",
"params": [[3, 2]],
"properties": {
"name": "A (A')",
"color": "#5583F2",
"size": 4,
"withLabel": true
}
},
{
"type": "point",
"params": [[2, 4]],
"properties": {
"name": "B",
"color": "#8C55F2",
"size": 3,
"withLabel": true
}
},
{
"type": "point",
"params": [[6, 3]],
"properties": {
"name": "C",
"color": "#8C55F2",
"size": 3,
"withLabel": true
}
},
{
"type": "point",
"params": [[1, 0]],
"properties": {
"name": "D",
"color": "#8C55F2",
"size": 3,
"withLabel": true
}
}
]
}
</plot>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Geometry",
"Coordinate Transformations"
]
}
</post_analysis>

Answer:

<pre_analysis>
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"step_cot",
"explain_then_answer",
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],
"needs_drawing": true,
"concepts_used": [
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],
"new_concepts": [
"Reflection Across Vertical Lines"
],
"current_concepts": [
"Coordinate Transformations",
"Reflection Across Vertical Lines"
]
}
</pre_analysis>

<reasoning>

Identify coordinates of point A

\[
A = (3, 2)
\]

Apply reflection rule across vertical line

\[

$$\begin{aligned} x' &= 2h - x \\ x' &= 2(3) - 3 = 3 \\ y' &= y = 2 \end{aligned}$$

\]

State coordinates of reflected point

\[
A' = (3, 2)
\]
</reasoning>

<answer>
\((3, 2)\)
</answer>

<plot>
{
"elements": [
{
"type": "line",
"params": [[3, -3], [3, 7]],
"properties": {
"name": "x = 3",
"strokeColor": "#F2557F",
"strokeWidth": 2,
"dash": 2,
"withLabel": true
}
},
{
"type": "polygon",
"params": [[3, 2], [2, 4], [6, 3], [1, 0]],
"properties": {
"fillColor": "rgba(140, 85, 242, 0.15)",
"strokeColor": "#8C55F2",
"strokeWidth": 2
}
},
{
"type": "point",
"params": [[3, 2]],
"properties": {
"name": "A (A')",
"color": "#5583F2",
"size": 4,
"withLabel": true
}
},
{
"type": "point",
"params": [[2, 4]],
"properties": {
"name": "B",
"color": "#8C55F2",
"size": 3,
"withLabel": true
}
},
{
"type": "point",
"params": [[6, 3]],
"properties": {
"name": "C",
"color": "#8C55F2",
"size": 3,
"withLabel": true
}
},
{
"type": "point",
"params": [[1, 0]],
"properties": {
"name": "D",
"color": "#8C55F2",
"size": 3,
"withLabel": true
}
}
]
}
</plot>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Geometry",
"Coordinate Transformations"
]
}
</post_analysis>