Question

which segment is the image of \\(\overline{ab}\\) reflected across the line \\(y = -2\\)?

Explanation:

<pre_analysis>
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"concepts_used": [
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"Translation Rule"
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"new_concepts": [
"Reflection Across Horizontal Lines"
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<reasoning>

Identify coordinates of the original segment

Using the Coordinate Transformations knowledge point
\[

$$\begin{aligned} &A = (2, -7)\\ &B = (8, -3) \end{aligned}$$

\]

Apply reflection rule across a horizontal line

Using the Coordinate Transformations and Translation Rule knowledge points
\[

$$\begin{aligned} &(x, y) \to (x, 2k - y) \quad \text{for } y = k\\ &k = -2\\ &y' = 2(-2) - y = -4 - y \end{aligned}$$

\]

Calculate the reflected coordinates

Using the Coordinate Transformations knowledge point
\[

$$\begin{aligned} &A'(x_A, y_A') = (2, -4 - (-7)) = (2, 3)\\ &B'(x_B, y_B') = (8, -4 - (-3)) = (8, -1) \end{aligned}$$

\]

Match with the segments on the graph

Using the Coordinate Transformations knowledge point
\[

$$\begin{aligned} &G = (2, 3)\\ &H = (8, -1)\\ &\text{Segment } GH \text{ connects } (2, 3) \text{ and } (8, -1) \end{aligned}$$

\]
</reasoning>

<answer>
\(\overline{GH}\)
</answer>

<plot>
{
"elements": [
{
"type": "line",
"params": [[-10, -2], [10, -2]],
"properties": {
"name": "y = -2",
"strokeColor": "#F2557F",
"strokeWidth": 2,
"dash": 2,
"withLabel": true,
"label": {
"position": "right"
}
}
},
{
"type": "segment",
"params": [[2, -7], [8, -3]],
"properties": {
"name": "AB",
"strokeColor": "#5583F2",
"strokeWidth": 3,
"withLabel": true
}
},
{
"type": "segment",
"params": [[2, 3], [8, -1]],
"properties": {
"name": "GH",
"strokeColor": "#8C55F2",
"strokeWidth": 3,
"withLabel": true
}
},
{
"type": "point",
"params": [[2, -7]],
"properties": {
"name": "A",
"color": "#5583F2"
}
},
{
"type": "point",
"params": [[8, -3]],
"properties": {
"name": "B",
"color": "#5583F2"
}
},
{
"type": "point",
"params": [[2, 3]],
"properties": {
"name": "G",
"color": "#8C55F2"
}
},
{
"type": "point",
"params": [[8, -1]],
"properties": {
"name": "H",
"color": "#8C55F2"
}
}
]
}
</plot>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Geometry",
"Reflection Across Horizontal Lines"
]
}
</post_analysis>

Answer:

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"concepts_used": [
"Coordinate Transformations",
"Translation Rule"
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"new_concepts": [
"Reflection Across Horizontal Lines"
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"Coordinate Transformations",
"Translation Rule",
"Reflection Across Horizontal Lines"
]
}
</pre_analysis>

<reasoning>

Identify coordinates of the original segment

Using the Coordinate Transformations knowledge point
\[

$$\begin{aligned} &A = (2, -7)\\ &B = (8, -3) \end{aligned}$$

\]

Apply reflection rule across a horizontal line

Using the Coordinate Transformations and Translation Rule knowledge points
\[

$$\begin{aligned} &(x, y) \to (x, 2k - y) \quad \text{for } y = k\\ &k = -2\\ &y' = 2(-2) - y = -4 - y \end{aligned}$$

\]

Calculate the reflected coordinates

Using the Coordinate Transformations knowledge point
\[

$$\begin{aligned} &A'(x_A, y_A') = (2, -4 - (-7)) = (2, 3)\\ &B'(x_B, y_B') = (8, -4 - (-3)) = (8, -1) \end{aligned}$$

\]

Match with the segments on the graph

Using the Coordinate Transformations knowledge point
\[

$$\begin{aligned} &G = (2, 3)\\ &H = (8, -1)\\ &\text{Segment } GH \text{ connects } (2, 3) \text{ and } (8, -1) \end{aligned}$$

\]
</reasoning>

<answer>
\(\overline{GH}\)
</answer>

<plot>
{
"elements": [
{
"type": "line",
"params": [[-10, -2], [10, -2]],
"properties": {
"name": "y = -2",
"strokeColor": "#F2557F",
"strokeWidth": 2,
"dash": 2,
"withLabel": true,
"label": {
"position": "right"
}
}
},
{
"type": "segment",
"params": [[2, -7], [8, -3]],
"properties": {
"name": "AB",
"strokeColor": "#5583F2",
"strokeWidth": 3,
"withLabel": true
}
},
{
"type": "segment",
"params": [[2, 3], [8, -1]],
"properties": {
"name": "GH",
"strokeColor": "#8C55F2",
"strokeWidth": 3,
"withLabel": true
}
},
{
"type": "point",
"params": [[2, -7]],
"properties": {
"name": "A",
"color": "#5583F2"
}
},
{
"type": "point",
"params": [[8, -3]],
"properties": {
"name": "B",
"color": "#5583F2"
}
},
{
"type": "point",
"params": [[2, 3]],
"properties": {
"name": "G",
"color": "#8C55F2"
}
},
{
"type": "point",
"params": [[8, -1]],
"properties": {
"name": "H",
"color": "#8C55F2"
}
}
]
}
</plot>

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"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Geometry",
"Reflection Across Horizontal Lines"
]
}
</post_analysis>