Question

missys rotation maps point \\(k(17, -12)\\) to \\(k(12, 17)\\). which describes the rotation?

\\(270^\circ\\) counterclockwise rotation
\\(90^\circ\\) counterclockwise rotation
\\(90^\circ\\) clockwise rotation
\\(180^\circ\\) rotation

Explanation:

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

<reasoning>

Identify coordinates

We are given the original point \(K(17, -12)\) and its image \(K'(12, 17)\) after a rotation about the origin. Let the coordinates of \(K\) be \((x, y) = (17, -12)\).

Analyze transformation

Using the Rotation Rules concept, we compare the coordinates of \(K(x, y)\) and \(K'(x', y')\):

• \(x = 17\)
• \(y = -12\)
• \(x' = 12 = -y\)
• \(y' = 17 = x\)

Thus, the mapping is \((x, y)
ightarrow (-y, x)\).

Determine rotation

Using the Rotation Rules concept, we identify the standard coordinate transformation rules about the origin:

• A \(90^\circ\) counterclockwise rotation maps \((x, y)

ightarrow (-y, x)\).

• A \(90^\circ\) clockwise rotation maps \((x, y)

ightarrow (y, -x)\).

• A \(180^\circ\) rotation maps \((x, y)

ightarrow (-x, -y)\).

• A \(270^\circ\) counterclockwise rotation maps \((x, y)

ightarrow (y, -x)\).

Since our mapping is \((x, y)
ightarrow (-y, x)\), this represents a \(90^\circ\) counterclockwise rotation.
</reasoning>

<answer>
<mcq-option>(A) \(270^\circ\) counterclockwise rotation</mcq-option>
<mcq-correct>(B) \(90^\circ\) counterclockwise rotation</mcq-correct>
<mcq-option>(C) \(90^\circ\) clockwise rotation</mcq-option>
<mcq-option>(D) \(180^\circ\) rotation</mcq-option>
</answer>

<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"
],
"needs_drawing": false,
"concepts_used": [
"Rotation Rules"
],
"new_concepts": [],
"current_concepts": [
"Rotation Rules",
"Coordinate Transformations"
]
}
</pre_analysis>

<reasoning>

Identify coordinates

We are given the original point \(K(17, -12)\) and its image \(K'(12, 17)\) after a rotation about the origin. Let the coordinates of \(K\) be \((x, y) = (17, -12)\).

Analyze transformation

Using the Rotation Rules concept, we compare the coordinates of \(K(x, y)\) and \(K'(x', y')\):

• \(x = 17\)
• \(y = -12\)
• \(x' = 12 = -y\)
• \(y' = 17 = x\)

Thus, the mapping is \((x, y)
ightarrow (-y, x)\).

Determine rotation

Using the Rotation Rules concept, we identify the standard coordinate transformation rules about the origin:

• A \(90^\circ\) counterclockwise rotation maps \((x, y)

ightarrow (-y, x)\).

• A \(90^\circ\) clockwise rotation maps \((x, y)

ightarrow (y, -x)\).

• A \(180^\circ\) rotation maps \((x, y)

ightarrow (-x, -y)\).

• A \(270^\circ\) counterclockwise rotation maps \((x, y)

ightarrow (y, -x)\).

Since our mapping is \((x, y)
ightarrow (-y, x)\), this represents a \(90^\circ\) counterclockwise rotation.
</reasoning>

<answer>
<mcq-option>(A) \(270^\circ\) counterclockwise rotation</mcq-option>
<mcq-correct>(B) \(90^\circ\) counterclockwise rotation</mcq-correct>
<mcq-option>(C) \(90^\circ\) clockwise rotation</mcq-option>
<mcq-option>(D) \(180^\circ\) rotation</mcq-option>
</answer>

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