Question

part b
list the coordinates of the vertices of the rotated rectangle.
enter the correct coordinates in the boxes.
(□,□), (□,□), (□,□), (□,□)

Explanation:

To solve for the coordinates of the vertices of the rotated rectangle, we typically need the original coordinates of the rectangle's vertices and the angle of rotation (or other rotation details). Since this information is not provided in the current problem statement, we can't proceed with the calculation. However, if we assume a common scenario (e.g., rotating a rectangle with vertices \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\), \((x_4,y_4)\) by a certain angle \(\theta\) around a point), the general steps would be:

Step 1: Recall the rotation formula

The rotation of a point \((x,y)\) around the origin by an angle \(\theta\) (counterclockwise) is given by:
\[

$$\begin{pmatrix} x' \\ y' \end{pmatrix}$$

=

$$\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$

$$\begin{pmatrix} x \\ y \end{pmatrix}$$

\]
which simplifies to:
\[
x' = x\cos\theta - y\sin\theta
\]
\[
y' = x\sin\theta + y\cos\theta
\]

Step 2: Apply the formula to each vertex

For each vertex \((x_i,y_i)\) of the original rectangle, calculate the new coordinates \((x_i',y_i')\) using the rotation formula.

Step 3: List the new coordinates

Once we have calculated the new coordinates for all four vertices, we list them as the vertices of the rotated rectangle.

Since the original coordinates and rotation details are missing, we can't provide the specific coordinates. Please provide the original coordinates of the rectangle's vertices and the rotation angle (or other rotation information) to get the final answer.

Answer:

To solve for the coordinates of the vertices of the rotated rectangle, we typically need the original coordinates of the rectangle's vertices and the angle of rotation (or other rotation details). Since this information is not provided in the current problem statement, we can't proceed with the calculation. However, if we assume a common scenario (e.g., rotating a rectangle with vertices \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\), \((x_4,y_4)\) by a certain angle \(\theta\) around a point), the general steps would be:

Step 1: Recall the rotation formula

The rotation of a point \((x,y)\) around the origin by an angle \(\theta\) (counterclockwise) is given by:
\[

$$\begin{pmatrix} x' \\ y' \end{pmatrix}$$

=

$$\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$

$$\begin{pmatrix} x \\ y \end{pmatrix}$$

\]
which simplifies to:
\[
x' = x\cos\theta - y\sin\theta
\]
\[
y' = x\sin\theta + y\cos\theta
\]

Step 2: Apply the formula to each vertex

For each vertex \((x_i,y_i)\) of the original rectangle, calculate the new coordinates \((x_i',y_i')\) using the rotation formula.

Step 3: List the new coordinates

Once we have calculated the new coordinates for all four vertices, we list them as the vertices of the rotated rectangle.

Since the original coordinates and rotation details are missing, we can't provide the specific coordinates. Please provide the original coordinates of the rectangle's vertices and the rotation angle (or other rotation information) to get the final answer.