Question

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)270 degrees (counterclockwise)270 degrees (counterclockwise)−90 degrees (clockwise)negative 90 degrees (clockwise)−180 degrees (clockwise) or 180 degrees (counterclockwise)

Answer 1

To determine the degree and direction of rotation for the point P(−4,10) to the point P′(−10,−4), we can use the rotation mapping formulas.

1. Identify coordinates:

   • Original point \( P(x, y) = (-4, 10) \)
   • Rotated point \( P'(x', y') = (-10, -4) \)

2. Check the possible rotation formulas. The rotation formulas involving degrees are:

   • For 90 degrees counterclockwise: \[ (x', y') = (-y, x) \]
   • For 180 degrees (both directions): \[ (x', y') = (-x, -y) \]
   • For 270 degrees counterclockwise (or 90 degrees clockwise): \[ (x', y') = (y, -x) \]

3. Test a 90-degree counterclockwise rotation: \[ P \text{ becomes } (-y, x) = (-10, -4) \] This does not match P′.

4. Test a 180-degree rotation: \[ P \text{ becomes } (-x, -y) = (4, -10) \] This does not match P′.

5. Test a 270-degree counterclockwise rotation (or 90 degrees clockwise): \[ P \text{ becomes } (y, -x) = (10, 4) \] This does not match P′ as well.

6. Test a 90-degree clockwise rotation (which is the same as 270 degrees counterclockwise): \[ P \text{ becomes } (-y, x) = (-10, -4) \] This does match P′.

Thus, the point \( P(−4, 10) \) is rotated to \( P' (−10, −4) \) through a 90 degrees clockwise rotation.

So the answer is: −90 degrees (clockwise).