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

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

Accepted Answer

{ "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" ] } ## 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 $$ \text{The transformation matches a } 90^\circ \text{ counter-clockwise rotation.} $$

counter-clockwise

horizontally

symmetrical

clockwise

{ "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)" } } ] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Rotation Rules" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze the given coordinates

Test the counter-clockwise rotation rule

Test the clockwise rotation rule

Determine the direction of rotation

Answer: