Question

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

Given that point \( Q(-4, -9) \) is rotated to point \( Q'(-9, 4) \), we can determine the center of rotation by using the midpoint of the line segment \( QQ' \).

### Step 1: Find the Center of Rotation

The midpoint \( M \) of points \( Q \) and \( Q' \) can be calculated as follows:

\[
M = \left(\frac{x_Q + x_{Q'}}{2}, \frac{y_Q + y_{Q'}}{2}\right)
\]

Substituting the coordinates of \( Q \) and \( Q' \):

\[
M = \left(\frac{-4 + (-9)}{2}, \frac{-9 + 4}{2}\right) = \left(\frac{-13}{2}, \frac{-5}{2}\right) = \left(-6.5, -2.5\right)
\]

### Step 2: Determine the Rotation Angle

Next, we need to calculate the angle of rotation. The vector from \( M \) to \( Q \) and the vector from \( M \) to \( Q' \) will provide us with the necessary information:

\[
\text{Vector } MQ = \left(-4 - (-6.5), -9 - (-2.5)\right) = \left(2.5, -6.5\right)
\]
\[
\text{Vector } MQ' = \left(-9 - (-6.5), 4 - (-2.5)\right) = \left(-2.5, 6.5\right)
\]

### Step 3: Rotate Points P and R

The rotation transformation around a point involves shifting the points to the origin, applying the rotation, and then shifting them back.

Using the rotation of \( 90^\circ \) as we can see from the vectors (the rotation from \( \left(2.5, -6.5\right) \) to \( \left(-2.5, 6.5\right) \) indicates a quarter turn counterclockwise in the Cartesian plane), we transform the other points.

### Rotation Function

For a point \( (x, y) \) rotated by \( 90^\circ \) counterclockwise around the center \( (h, k) \):

1. Translate point to origin:
\[
x' = x - h
\]
\[
y' = y - k
\]

2. Rotate:
\[
x'' = -y'
\]
\[
y'' = x'
\]

3. Translate back:
\[
x_{\text{final}} = x'' + h
\]
\[
y_{\text{final}} = y'' + k
\]

### Apply to \( P(-4, 0) \):

Substituting into the transformation steps:

1. Translate:
\[
x' = -4 - (-6.5) = 2.5
\]
\[
y' = 0 - (-2.5) = 2.5
\]

2. Rotate:
\[
x'' = -2.5
\]
\[
y'' = 2.5
\]

3. Translate back:
\[
x_{\text{final}} = -2.5 + (-6.5) = -9
\]
\[
y_{\text{final}} = 2.5 + (-2.5) = 0
\]

So, the new coordinates for vertex \( P \) should be \( P'(-9, 0) \).

### Apply to \( R(4, -4) \):

1. Translate:
\[
x' = 4 - (-6.5) = 10.5
\]
\[
y' = -4 - (-2.5) = -1.5
\]

2. Rotate:
\[
x'' = 1.5
\]
\[
y'' = 10.5
\]

3. Translate back:
\[
x_{\text{final}} = 1.5 + (-6.5) = -5
\]
\[
y_{\text{final}} = 10.5 + (-2.5) = 8
\]

So, the new coordinates for vertex \( R \) should be \( R'(-5, 8) \).

### Final Results:
- \( P' \) is at \( (-9, 0) \).
- \( R' \) is at \( (-5, 8) \).

Thus, the positions of the images of vertices \( P \) and \( R \) after rotation are:
- \( P'(-9, 0) \)
- \( R'(-5, 8) \)