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

Answer 1

To determine the angle of rotation that takes point \( P(-4, 10) \) to point \( P'(-10, -4) \), we can analyze their positions in relation to the origin.

1. **Identify the original and new coordinates**:
- \( P(x_1, y_1) = (-4, 10) \)
- \( P'(x_2, y_2) = (-10, -4) \)

2. **Determine the direction and angle of rotation**:
To find the angle of rotation, we will use the formula for rotation of a point around the origin. The general rotation transformation is given by:

\[
\begin{pmatrix}
x' \\
y'
\end{pmatrix}
=
\begin{pmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
\]

We will check the coordinates:

After rotation by \( \theta \):
- For point \( P(-4, 10) \):

\[
x' = -4 \cos \theta + 10 \sin \theta
\]
\[
y' = -4 \sin \theta + 10 \cos \theta
\]

This should equal \( P'(-10, -4) \):
\[
-4 \cos \theta + 10 \sin \theta = -10 \quad (1)
\]
\[
-4 \sin \theta + 10 \cos \theta = -4 \quad (2)
\]

3. **Substituting values** and calculating the angle \(\theta\):

Let's analyze the coordinates geometrically:
- The original point \( P(-4, 10) \) is in the second quadrant.
- The new point \( P'(-10, -4) \) is in the third quadrant.

### Finding the angle of rotation:

We can calculate the angle from the origin to both points.

**For point P**:

\[
\tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{10}{-4}\right) = \tan^{-1}\left(-\frac{5}{2}\right) \approx 180 - \theta \quad (\text{since it's in the second quadrant})
\]

So, the angle for \( P \) is approximately \( 180 - \theta \).

**For point P'**:

\[
\tan^{-1}\left(\frac{-4}{-10}\right) = \tan^{-1}\left(\frac{2}{5}\right) \text{ (since both coordinates are negative, it's in the third quadrant)}.
\]

Thus, the angle corresponding to \( P' \) is \( 180 + \tan^{-1}\left(\frac{2}{5}\right)\).

### Calculating rotation angle:

The angle moved from \( P \) to \( P' \) can be computed.

We find:

1. **From initial approximate angle values**, \( P \approx 180^\circ - \theta \)
2. \( P' \approx 180^\circ + \tan^{-1}\left(\frac{2}{5}\right) + 180^\circ \)

### Verifying angle shift:
Calculating the shift gives you \( 180^\circ \).

**Final Result**:

The rotation from \( P \) to \( P' \) can be confirmed as \( 180^\circ \) counterclockwise or \( -180^\circ \) clockwise.

### Conclusion:
Thus, the answer is:

**180 degrees (counterclockwise)** or **−180 degrees (clockwise)**.