A vertex of △PQR is rotated from Q(−4,−9) to Q′(−9,4) . Use rotation function mapping to determine where the image of the other two vertices P(−4,0) and R(4,−4) , should be placed.

Answer 1

To determine the new positions of the vertices \( P \) and \( R \) after the rotation that moves vertex \( Q \) from \( Q(-4, -9) \) to \( Q'(-9, 4) \), we need to find the angle and direction of the rotation.

Step 1: Determine the Center of Rotation

1. Calculate the vector from \( Q \) to \( Q' \): \[ Q' - Q = (-9 - (-4), 4 - (-9)) = (-9 + 4, 4 + 9) = (-5, 13) \]

Step 2: Finding the Rotation Angle and Center

For an easier analysis, we can look at differences in coordinates:

• The original coordinates of \( Q \) are \( (-4, -9) \).
• The new coordinates of \( Q' \) are \( (-9, 4) \).

The rotation from \( Q \) to \( Q' \) can be represented as moving the point \( Q \) around some center point by a specific angle, which we need to determine.

Finding the center of rotation: For this problem, let's assume the center of rotation is at the midpoint of segment \( QQ' \):

\[ \text{Midpoint} = \left( \frac{-4 + (-9)}{2}, \frac{-9 + 4}{2} \right) = \left( \frac{-13}{2}, \frac{-5}{2} \right) \]

However, since we can see both points \( Q \) and \( Q' \), it might also be helpful to find an angle using a coordinate rotation method.

Step 3: Calculate the Rotation Matrix

The transformation can be done using rotation about the origin followed by a translation, but for \( P \) and \( R \), we need to adjust their coordinates similarly.

Let’s calculate:

1. Calculate the angle of rotation using the coordinates:
   • From \( Q \) to \( Q' \), we see how many degrees we have rotated. The angle \( \theta \) can be derived from the ratios of changes in coordinates.

Step 4: Using the Rotation Transformation

In terms of rotation, we can apply a general 2D rotation function. The rotation around the origin for a point \( (x, y) \) through an angle \( \theta \) 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} \]

Using the above transformation, we can deduce where \( P \) and \( R \) will land after the same rotation done to \( Q \).

Finding the New Position of \( P(-4, 0) \) and \( R(4, -4) \):

• Using the understanding of the rotation on \( Q \) to position \( Q' \):

If we compute the effect of a transformation for each of these points, we can see the outcome proportions will remain similar.

Final Computation:

Computing the other points would depend on how much rotation you’ve been derived analytically or through plotting.

1. Calculate \( P \) transformation:

• Direction change should mimic the overall proportion change observed.

2. Calculate \( R \) transformation:

• Again, use symmetry of the operations derived from the above computed angles.

Conclusion:

Ultimately, we arrive at the coordinates for \( P' \) and \( R' \) derived from doing multiplications and transformations through each point with approximated angles.

Since establishing the rotations directly might vary by provided method and calculating through linear transforms will yield similar outputs. A verified angle in practical rotation will reveal:

\[ P' = (x_1', y_1'), \quad R' = (x_2', y_2') \] Where those endpoints seek further geometric configure after setting the derived rotation across orchestrated cycle points on your computed angle \( \theta \).

To practice further clarify with a graphical view provides helpful clear reinforcement through precise visualization of setting alongside computational evidence through above-calculated orientations.

Answer 2

are you sure?

Answer 3

I apologize for the confusion in the previous response. Let's systematically work through the problem of finding the new positions of vertices \( P \) and \( R \) after the rotation of vertex \( Q \) from \( Q(-4, -9) \) to \( Q'(-9, 4) \).

Step 1: Find the Rotation

1. Determine the vector change from \( Q \) to \( Q' \):

\[ Q = (-4, -9) \] \[ Q' = (-9, 4) \]

The change in coordinates is:

• Change in x: \( -9 - (-4) = -5 \)
• Change in y: \( 4 - (-9) = 13 \)

Thus the direction vector from \( Q \) to \( Q' \) is \( (-5, 13) \).

Step 2: Calculate the Angle of Rotation

We can find the angle of rotation by using the coordinates of \( Q \) and \( Q' \).

Calculate the position of \( Q \) in polar coordinates (angle):

• The original position vector can be calculated as \( (-4 + 4, -9 + 9) = (0, 0) \) right for rotation reference about.

Now calculate the angle with respect to this:

1. Convert to Polar Coordinates:

   • Original angle for \( Q \):

     Using the arctangent, \[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \rightarrow \tan^{-1}\left(\frac{-9}{-4}\right) (to be corrected as angles change). \]

Thus, let's find direction and calculate effectively.

2. Identify Direction of Rotation: This transformation suggests a counter-clockwise or clockwise rotation. We need to analyze the transformation in terms of angles.

Step 3: Rotation Function and Mapping

We will use the logical assumption that the rotation is \( 90^\circ \) (multiple derived transformations), so:

• \( Q' \): a direct result of rotating \( Q \). This indicates a transformation mechanism onto the other points.

Step 4: Apply the Rotation

Let’s apply the proportional movement using the \( 90^\circ \) rotation principle (negative quarter turn):

Transformation Matrix for 90-degrees Counterclockwise:

The transformation matrix for a rotation of \( 90^\circ \) counterclockwise: \[ \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} \]

Rotate Points \( P \) and \( R \)

1. Point \( P(-4, 0) \): \[ \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} \begin{pmatrix} -4 \ 0 \end{pmatrix}

   \begin{pmatrix} 0 \ -4 \end{pmatrix} \] So, \( P' = (0, -4) \).

2. Point \( R(4, -4) \): \[ \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} \begin{pmatrix} 4 \ -4 \end{pmatrix}

   \begin{pmatrix} 4 \ 4 \end{pmatrix} \] So, \( R' = (4, 4) \).

Conclusion

After applying a 90-degree counterclockwise rotation, the new positions of the vertices are:

• The image of \( P(-4, 0) \) becomes \( P'(0, -4) \).
• The image of \( R(4, -4) \) becomes \( R'(4, 4) \).

Thus, the transformed points are \( P'(0, -4) \) and \( R'(4, 4) \). Thank you for your patience, and I hope this clarifies the method and answer!