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.(1 point)
Responses
P′(0,−4)
and R′(−4,4)
upper P prime left parenthesis 0 comma negative 4 right parenthesis and upper R left parenthesis negative 4 comma 4 right parenthesis
P′(0,4)
and R′(−4,4)
upper P prime left parenthesis 0 comma 4 right parenthesis and upper R left parenthesis negative 4 comma 4 right parenthesis
P′(0,4)
and R′(−4,−4)
upper P prime left parenthesis 0 comma 4 right parenthesis and upper R left parenthesis negative 4 comma negative 4 right parenthesis
P′(0,−4)
and R′(−4,−4)
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.(1 point)
Responses
P′(0,−4)
and R′(−4,4)
upper P prime left parenthesis 0 comma negative 4 right parenthesis and upper R left parenthesis negative 4 comma 4 right parenthesis
P′(0,4)
and R′(−4,4)
upper P prime left parenthesis 0 comma 4 right parenthesis and upper R left parenthesis negative 4 comma 4 right parenthesis
P′(0,4)
and R′(−4,−4)
upper P prime left parenthesis 0 comma 4 right parenthesis and upper R left parenthesis negative 4 comma negative 4 right parenthesis
P′(0,−4)
and R′(−4,−4)
Answers
GPT-4o mini
To determine the images of the other vertices \( P \) and \( R \) after the rotation that moved \( Q(-4, -9) \) to \( Q'(-9, 4) \), we first need to understand the rotation transformation.
1. **Identifying the rotation angle and center**:
- Determine the center of rotation. The midpoint between \( Q \) and \( Q' \) can help us find that.
- The midpoint \( M \) is given by:
\[
M = \left( \frac{-4 + -9}{2}, \frac{-9 + 4}{2} \right) = \left( \frac{-13}{2}, \frac{-5}{2} \right)
\]
- The angle of rotation can be calculated by using the coordinates of points before and after the rotation.
2. **Calculate the vector from \( Q \) to \( Q' \)**:
- The vector from \( Q \) to \( Q' \) is:
\[
Q' - Q = (-9 - (-4), 4 - (-9)) = (-5, 13)
\]
- This vector gives us the direction of the rotation.
3. **Apply the same transformation to \( P \) and \( R \)**:
- To perform the same rotation on the other vertices, we find how much the coordinates of \( P \) and \( R \) would change under this transformation.
4. **Rotation by \( 90^\circ \) counter-clockwise** around an arbitrary point follows the rule: If a point \( (x, y) \) is rotated around a point \( (h, k) \):
- New coordinates \( (x', y') \) can be computed as:
\[
(x', y') = (h - (y - k), k + (x - h))
\]
Assuming the rotation is \( 90^\circ \) counter-clockwise around the calculated midpoint \( M \) (which we need to adjust to be the origin for simplicity), we will apply to each point:
For \( P(-4, 0) \):
- Shift to origin: Translate to \( (4, 0) \)
- Apply rotation:
\[
P' = (0, 4) \text{ (after rotating)}
\]
- Translate back: \( P' = (0, 4) \)
For \( R(4, -4) \):
- Shift to origin: \( (-4, -4) \)
- Apply rotation:
\[
R' = (-4, -4) \text{ (does not change)}
\]
- Translate back: \( R' = (-4, -4) \)
Thus the new coordinates for \( P' \) and \( R' \) after rotation should be:
\[
P' = (0, 4) \quad \text{and} \quad R' = (-4, -4)
\]
From the options given, the correct pair is:
**P′(0, 4) and R′(−4, −4)**.
1. **Identifying the rotation angle and center**:
- Determine the center of rotation. The midpoint between \( Q \) and \( Q' \) can help us find that.
- The midpoint \( M \) is given by:
\[
M = \left( \frac{-4 + -9}{2}, \frac{-9 + 4}{2} \right) = \left( \frac{-13}{2}, \frac{-5}{2} \right)
\]
- The angle of rotation can be calculated by using the coordinates of points before and after the rotation.
2. **Calculate the vector from \( Q \) to \( Q' \)**:
- The vector from \( Q \) to \( Q' \) is:
\[
Q' - Q = (-9 - (-4), 4 - (-9)) = (-5, 13)
\]
- This vector gives us the direction of the rotation.
3. **Apply the same transformation to \( P \) and \( R \)**:
- To perform the same rotation on the other vertices, we find how much the coordinates of \( P \) and \( R \) would change under this transformation.
4. **Rotation by \( 90^\circ \) counter-clockwise** around an arbitrary point follows the rule: If a point \( (x, y) \) is rotated around a point \( (h, k) \):
- New coordinates \( (x', y') \) can be computed as:
\[
(x', y') = (h - (y - k), k + (x - h))
\]
Assuming the rotation is \( 90^\circ \) counter-clockwise around the calculated midpoint \( M \) (which we need to adjust to be the origin for simplicity), we will apply to each point:
For \( P(-4, 0) \):
- Shift to origin: Translate to \( (4, 0) \)
- Apply rotation:
\[
P' = (0, 4) \text{ (after rotating)}
\]
- Translate back: \( P' = (0, 4) \)
For \( R(4, -4) \):
- Shift to origin: \( (-4, -4) \)
- Apply rotation:
\[
R' = (-4, -4) \text{ (does not change)}
\]
- Translate back: \( R' = (-4, -4) \)
Thus the new coordinates for \( P' \) and \( R' \) after rotation should be:
\[
P' = (0, 4) \quad \text{and} \quad R' = (-4, -4)
\]
From the options given, the correct pair is:
**P′(0, 4) and R′(−4, −4)**.