Question

what is the mapping rule for a 180 degree rotation about the origin?

(x, y) \
ightarrow (-y, x)
(x, y) \
ightarrow (-y, -x)
(x, y) \
ightarrow (-x, -y)
(x, y) \
ightarrow (x, -y)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
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"concepts_used": [
"Coordinate Transformations"
],
"new_concepts": [
"Rotation Rules"
],
"current_concepts": [
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"Rotation Rules"
]
}
</pre_analysis>

<reasoning>

Identify the transformation type

\[
\text{Rotation of } 180^\circ \text{ about the origin } (0,0).
\]

Apply the coordinate rule

\[
(x, y)
ightarrow (-x, -y)
\]

Match with the given options

\[
\text{Option 3: } (x, y)
ightarrow (-x, -y)
\]
</reasoning>

<answer>
<mcq-option>(A) \((x, y)
ightarrow (-y, x)\)</mcq-option>
<mcq-option>(B) \((x, y)
ightarrow (-y, -x)\)</mcq-option>
<mcq-correct>(C) \((x, y)
ightarrow (-x, -y)\)</mcq-correct>
<mcq-option>(D) \((x, y)
ightarrow (x, -y)\)</mcq-option>
</answer>

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

<reasoning>

Identify the transformation type

\[
\text{Rotation of } 180^\circ \text{ about the origin } (0,0).
\]

Apply the coordinate rule

\[
(x, y)
ightarrow (-x, -y)
\]

Match with the given options

\[
\text{Option 3: } (x, y)
ightarrow (-x, -y)
\]
</reasoning>

<answer>
<mcq-option>(A) \((x, y)
ightarrow (-y, x)\)</mcq-option>
<mcq-option>(B) \((x, y)
ightarrow (-y, -x)\)</mcq-option>
<mcq-correct>(C) \((x, y)
ightarrow (-x, -y)\)</mcq-correct>
<mcq-option>(D) \((x, y)
ightarrow (x, -y)\)</mcq-option>
</answer>

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