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.

Question

A. -90 degrees (clockwise)
B. -180 degrees (clockwise) or 180 degrees (counterclockwise)
C. 270 degrees (counterclockwise)
D. 90 degrees (counterclockwise)

Answer 1

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

For a point \( (x, y) \), the rotation formulas for a counterclockwise rotation by an angle θ are:

• \( (x', y') = (x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta) \)

We will identify which angle θ fits this transformation from P to P′.

Starting with \( P(−4, 10) \) and \( P'(-10, -4) \):

1. Compute the distances from the origin to both points:
   • Distance of P: \( \sqrt{(-4)^2 + 10^2} = \sqrt{16 + 100} = \sqrt{116} \)
   • Distance of P′: \( \sqrt{(-10)^2 + (-4)^2} = \sqrt{100 + 16} = \sqrt{116} \)

Both points are the same distance from the origin, confirming the rotation is valid.

2. We can find the equation resulting from rotation by checking the changes in coordinates:

   • From \( (−4, 10) \) to \( P′(−10, −4) \):
     • The new x-coordinate from the old is \( x' = -10 \)
     • The new y-coordinate from the old is \( y' = -4 \)

3. A useful approach is to visualize or calculate the angles:

   • The original point \( P(−4, 10) \) can be visualized in the second quadrant where x is negative and y is positive.
   • The resulting point \( P'(-10, -4) \) is in the third quadrant where both x and y are negative.

Next, we can use the angle values:

• The original angle \( P \) makes with the positive x-axis is roughly in the second quadrant (arctan(10/-4)), and the new angle \( P' \) makes with the positive x-axis is in the third quadrant.

To find the angle of rotation:

1. Calculate the angle of P: \[ \tan^{-1}(\frac{10}{-4}) = \tan^{-1}(-2.5) \]

2. The angle \( P' \) is: \[ \tan^{-1}(\frac{-4}{-10}) = \tan^{-1}(0.4) \]

Now we just need to find the equivalent angles based on the standard position of angles:

• Angles in the second quadrant are given a negative measure, and we often compute in terms of degrees in standard (0° to 360°):
• Determine rotation: Eyeballing the movement from each quadrant should suggest clockwise or counterclockwise rotation by \( 180° or -180° \).

Putting it all together, given that:

• \( P'(-10, -4) \) is a direct rotation implies the choices:
• Checking directions, we see that moving from the second quadrant toward the third quadrant exemplifies a -90° rotation (clockwise).

Thus, the correct match from the options provided is:

A. -90 degrees (clockwise)