Question

point g is rotated \\(90^{\circ}\\). the coordinate of the pre-image point g was \\((7, -5)\\), and its image g is at the coordinate \\((5, 7)\\). what is the direction of the rotation?

counter-clockwise
horizontally
symmetrical
clockwise

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": [
"Rotation Rules",
"Clockwise Rotation Angle"
],
"new_concepts": [],
"current_concepts": [
"Rotation Rules",
"Clockwise Rotation Angle",
"Coordinate Transformations"
]
}
</pre_analysis>

<reasoning>

Analyze the given coordinates

Using the Rotation Rules and Coordinate Transformations knowledge points
\[

$$\begin{aligned} &G = (x, y) = (7, -5)\\ &G' = (x', y') = (5, 7) \end{aligned}$$

\]

Test the counter-clockwise rotation rule

Using the Rotation Rules knowledge point
\[

$$\begin{aligned} &R_{90^\circ, \text{ counter-clockwise}}(x, y) = (-y, x)\\ &(-y, x) = (-(-5), 7) = (5, 7)\\ &(5, 7) = G' \end{aligned}$$

\]

Test the clockwise rotation rule

Using the Clockwise Rotation Angle and Rotation Rules knowledge points
\[

$$\begin{aligned} &R_{90^\circ, \text{ clockwise}}(x, y) = (y, -x)\\ &(y, -x) = (-5, -7) e G' \end{aligned}$$

\]

Determine the direction of rotation

Using the Rotation Rules knowledge point
\[
\text{The transformation matches a } 90^\circ \text{ counter-clockwise rotation.}
\]
</reasoning>

<answer>
<mcq-correct>counter-clockwise</mcq-correct>
<mcq-option>horizontally</mcq-option>
<mcq-option>symmetrical</mcq-option>
<mcq-option>clockwise</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[7, -5]],
"properties": {
"name": "G (7, -5)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[5, 7]],
"properties": {
"name": "G' (5, 7)",
"size": 4,
"color": "#F2557F",
"withLabel": true
}
},
{
"type": "point",
"params": [[0, 0]],
"properties": {
"name": "O (0, 0)",
"size": 3,
"color": "black",
"withLabel": true
}
},
{
"type": "segment",
"params": [[0, 0], [7, -5]],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 1,
"dash": 2
}
},
{
"type": "segment",
"params": [[0, 0], [5, 7]],
"properties": {
"strokeColor": "#F2557F",
"strokeWidth": 1,
"dash": 2
}
},
{
"type": "angle",
"params": [[7, -5], [0, 0], [5, 7]],
"properties": {
"radius": 1.5,
"strokeColor": "#5583F2",
"fillColor": "rgba(85, 131, 242, 0.2)"
}
}
]
}
</plot>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Rotation Rules"
]
}
</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": [
"Rotation Rules",
"Clockwise Rotation Angle"
],
"new_concepts": [],
"current_concepts": [
"Rotation Rules",
"Clockwise Rotation Angle",
"Coordinate Transformations"
]
}
</pre_analysis>

<reasoning>

Analyze the given coordinates

Using the Rotation Rules and Coordinate Transformations knowledge points
\[

$$\begin{aligned} &G = (x, y) = (7, -5)\\ &G' = (x', y') = (5, 7) \end{aligned}$$

\]

Test the counter-clockwise rotation rule

Using the Rotation Rules knowledge point
\[

$$\begin{aligned} &R_{90^\circ, \text{ counter-clockwise}}(x, y) = (-y, x)\\ &(-y, x) = (-(-5), 7) = (5, 7)\\ &(5, 7) = G' \end{aligned}$$

\]

Test the clockwise rotation rule

Using the Clockwise Rotation Angle and Rotation Rules knowledge points
\[

$$\begin{aligned} &R_{90^\circ, \text{ clockwise}}(x, y) = (y, -x)\\ &(y, -x) = (-5, -7) e G' \end{aligned}$$

\]

Determine the direction of rotation

Using the Rotation Rules knowledge point
\[
\text{The transformation matches a } 90^\circ \text{ counter-clockwise rotation.}
\]
</reasoning>

<answer>
<mcq-correct>counter-clockwise</mcq-correct>
<mcq-option>horizontally</mcq-option>
<mcq-option>symmetrical</mcq-option>
<mcq-option>clockwise</mcq-option>
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[7, -5]],
"properties": {
"name": "G (7, -5)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[5, 7]],
"properties": {
"name": "G' (5, 7)",
"size": 4,
"color": "#F2557F",
"withLabel": true
}
},
{
"type": "point",
"params": [[0, 0]],
"properties": {
"name": "O (0, 0)",
"size": 3,
"color": "black",
"withLabel": true
}
},
{
"type": "segment",
"params": [[0, 0], [7, -5]],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 1,
"dash": 2
}
},
{
"type": "segment",
"params": [[0, 0], [5, 7]],
"properties": {
"strokeColor": "#F2557F",
"strokeWidth": 1,
"dash": 2
}
},
{
"type": "angle",
"params": [[7, -5], [0, 0], [5, 7]],
"properties": {
"radius": 1.5,
"strokeColor": "#5583F2",
"fillColor": "rgba(85, 131, 242, 0.2)"
}
}
]
}
</plot>

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