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A point is rotated about the origin. Its coordinates are P(−4,10) and P′(−10,−4). Determine the degree of rotation and direction by applying rotation mapping.(1 point)Responses90 degrees (counterclockwise)90 degrees (counterclockwise)−90 degrees (clockwise)negative 90 degrees (clockwise)270 degrees (counterclockwise)270 degrees (counterclockwise)−180 degrees (clockwise) or 180 degrees (counterclockwise)

1 answer

To determine the degree and direction of rotation from point \( P(-4, 10) \) to point \( P'(-10, -4) \), we can apply the rotation mapping.

For a point \( (x, y) \), the standard rotation mappings about the origin are:

  • 90 degrees counterclockwise: \( (x, y) \rightarrow (-y, x) \)
  • 180 degrees (clockwise or counterclockwise): \( (x, y) \rightarrow (-x, -y) \)
  • 270 degrees counterclockwise (or equivalently, 90 degrees clockwise): \( (x, y) \rightarrow (y, -x) \)

Now let’s analyze the transformation that takes \( P(-4, 10) \) to \( P'(-10, -4) \):

  1. Starting point \( P(-4, 10) \):

    • After 90 degrees counterclockwise: \[ (-10, -4) \quad (this is P') \] Thus, applying the mapping confirms:

    \[ P(-4, 10) \rightarrow (-10, -4) \text{ after a rotation of } 90 \text{ degrees counterclockwise.} \]

Since the transformation from \( P \) to \( P' \) matches the 90 degrees counterclockwise mapping, the answer is:

90 degrees (counterclockwise).