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)
Responses

90 degrees (counterclockwise)
90 degrees (counterclockwise)

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

270 degrees (counterclockwise)
270 degrees (counterclockwise)

−90 degrees (clockwise)

Answer 1

To determine the degree of rotation and direction for the points \( P(-4, 10) \) and \( P'(-10, -4) \), we can use the rotation mapping of coordinates.

The general formulas for rotating a point \( (x, y) \) about the origin by \(\theta\) degrees counterclockwise are:

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

We are given:
- Initial point \( P(-4, 10) \)
- Rotated point \( P'(-10, -4) \)

To find the angle \(\theta\), we can examine the transformations of the coordinates. Let's find out what happens to \( P \).

1. **Calculate the original distance of point P from the origin**:
\[
d_P = \sqrt{(-4)^2 + (10)^2} = \sqrt{16 + 100} = \sqrt{116} \approx 10.77
\]

2. **Calculate the distance of point P' from the origin:**
\[
d_{P'} = \sqrt{(-10)^2 + (-4)^2} = \sqrt{100 + 16} = \sqrt{116} \approx 10.77
\]

Both points \( P \) and \( P' \) are the same distance from the origin, confirming that this is a rotation around the origin.

3. **Next, to find an angle, we will consider the transformation of their coordinates.**

Original coordinates: \( P(-4, 10) \)

After rotation:
\[
P'(-10, -4)
\]

To determine the angle of rotation, look at the changes in their coordinates:

- The x-coordinate changes from \(-4\) to \(-10\)
- The y-coordinate changes from \(10\) to \(-4\)

4. **Geometric visualization:**
- Point \( P \) lies in the second quadrant (negative x, positive y).
- Point \( P' \) lies in the third quadrant (negative x, negative y).

This indicates that the rotation has moved the point from the second quadrant to the third quadrant - typically reflecting a rotation of \( 180^{\circ} \).

5. **Calculating for the exact angle:**

Since the coordinates have shifted from:
- \( x = -4 \) to \( x = -10 \) (which indicates a shift to negative on x)
- \( y = 10 \) to \( y = -4 \) (which indicates a shift to negative on y)

This confirms that the rotation is:

\[
-180^\circ \text{ (clockwise) or } 180^\circ \text{ (counterclockwise)}
\]

So the answer is:
\[
\text{−180 degrees (clockwise) or 180 degrees (counterclockwise)}
\]