To determine which transformation occurred to map point \( Q(-3, -5) \) to point \( Q'(5, -3) \), we can analyze the coordinates and the nature of the transformations.
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Check each transformation option:
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A. A 270-degree (counterclockwise) rotation:
- A 270-degree counterclockwise rotation changes the coordinates \( (x, y) \) to \( (y, -x) \).
- Applying this to \( Q(-3, -5) \): \[ (-5, 3) \quad \text{(not equal to } Q' (5, -3) \text{)} \]
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B. A reflection across the x-axis:
- Reflecting across the x-axis changes the coordinates \( (x, y) \) to \( (x, -y) \).
- Applying this to \( Q(-3, -5) \): \[ (-3, 5) \quad \text{(not equal to } Q' (5, -3) \text{)} \]
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C. A 90-degree (counterclockwise) rotation:
- A 90-degree counterclockwise rotation changes the coordinates \( (x, y) \) to \( (-y, x) \).
- Applying this to \( Q(-3, -5) \): \[ (5, -3) \quad \text{(which matches } Q' (5, -3) \text{)} \]
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D. A 180-degree rotation:
- A 180-degree rotation changes the coordinates \( (x, y) \) to \( (-x, -y) \).
- Applying this to \( Q(-3, -5) \): \[ (3, 5) \quad \text{(not equal to } Q' (5, -3) \text{)} \]
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After checking each option, the only transformation that correctly maps \( Q(-3, -5) \) to \( Q'(5, -3) \) is C. a 90-degree (counterclockwise) rotation.