Question
Triangle PQR is reflected over a vertical line of reflection to create triangle P′Q′R′ . What are the coordinates of point R′ ? (1 point) Responses (1,−3) left parenthesis 1 comma negative 3 right parenthesis (−2,7) left parenthesis negative 2 comma 7 right parenthesis (11,1) left parenthesis 11 comma 1 right parenthesis (−3,1)
Answers
Answer
R coordinates are, (-11,1)
Answer
R coordinates are, (-11,1) So, what is R' coordinates?
Answer
The coordinates of PQR are, (-8,9) (-12,7) (-11,1). P' coordinates are (-6,9)
Answer
The coordinates of PQR are, (-8,9) (-12,7) (-11,1)
Answered by
GPT-4o mini
To determine the coordinates of point \( R' \) after triangle \( PQR \) is reflected over a vertical line of reflection, we need to know the coordinates of point \( R \) and the line of reflection.
However, since no specific coordinates of point \( R \) and the vertical line of reflection have been provided in your question, we cannot definitively calculate the new coordinates of \( R' \).
If you can provide the coordinates for point \( R \) and the equation of the vertical line of reflection, I would be able to assist you in finding the coordinates of point \( R' \).
However, since no specific coordinates of point \( R \) and the vertical line of reflection have been provided in your question, we cannot definitively calculate the new coordinates of \( R' \).
If you can provide the coordinates for point \( R \) and the equation of the vertical line of reflection, I would be able to assist you in finding the coordinates of point \( R' \).
Answered by
GPT-4o mini
To determine the coordinates of point \( R' \) after triangle \( PQR \) is reflected over a vertical line of reflection, we first need to identify the vertical line of reflection.
However, since you have not specified the line of reflection, let's assume a general case:
1. **If the vertical line of reflection is given as \( x = a \):**
- The formula for reflection over a vertical line is:
\[
R' = (2a - x_R, y_R)
\]
- Here, \( (x_R, y_R) \) are the coordinates of point \( R \), which is \( (-11, 1) \).
2. **Example:**
- If we reflect over the line \( x = 0 \) (the y-axis):
\[
R' = (2(0) - (-11), 1) = (11, 1)
\]
3. **List of Possible Reflections:**
- If you provide specific values for \( a \), you can calculate \( R' \) accordingly.
Given the original point \( R \) at \( (-11, 1) \), the possible coordinates for \( R' \) can be calculated depending on the specific vertical line of reflection you are using.
Please provide the line of reflection for a definitive answer. Otherwise, from your earlier listed choices, if we consider a reflection across \( x = 0 \), then \( R' \) would be \( (11, 1) \).
However, since you have not specified the line of reflection, let's assume a general case:
1. **If the vertical line of reflection is given as \( x = a \):**
- The formula for reflection over a vertical line is:
\[
R' = (2a - x_R, y_R)
\]
- Here, \( (x_R, y_R) \) are the coordinates of point \( R \), which is \( (-11, 1) \).
2. **Example:**
- If we reflect over the line \( x = 0 \) (the y-axis):
\[
R' = (2(0) - (-11), 1) = (11, 1)
\]
3. **List of Possible Reflections:**
- If you provide specific values for \( a \), you can calculate \( R' \) accordingly.
Given the original point \( R \) at \( (-11, 1) \), the possible coordinates for \( R' \) can be calculated depending on the specific vertical line of reflection you are using.
Please provide the line of reflection for a definitive answer. Otherwise, from your earlier listed choices, if we consider a reflection across \( x = 0 \), then \( R' \) would be \( (11, 1) \).
Answered by
GPT-4o mini
To find the coordinates of point \( R' \) after reflecting point \( R(-11, 1) \) over a vertical line, we need to know the equation of the vertical line of reflection.
Let's consider a few options for vertical lines:
1. **If the vertical line of reflection is \( x = 0 \):**
\[
R' = (2(0) - (-11), 1) = (11, 1)
\]
2. **If the vertical line of reflection is \( x = -5 \):**
\[
R' = (2(-5) - (-11), 1) = (-10 + 11, 1) = (1, 1)
\]
3. **If the vertical line of reflection is \( x = -3 \):**
\[
R' = (2(-3) - (-11), 1) = (-6 + 11, 1) = (5, 1)
\]
For each specific vertical line of reflection, the resulting coordinates of \( R' \) will vary.
### Conclusion
To provide a precise answer for \( R' \)'s coordinates, please confirm the vertical line of reflection that triangle \( PQR \) is being reflected over. If the line was the y-axis (\( x = 0 \)), then \( R' = (11, 1) \).
Let's consider a few options for vertical lines:
1. **If the vertical line of reflection is \( x = 0 \):**
\[
R' = (2(0) - (-11), 1) = (11, 1)
\]
2. **If the vertical line of reflection is \( x = -5 \):**
\[
R' = (2(-5) - (-11), 1) = (-10 + 11, 1) = (1, 1)
\]
3. **If the vertical line of reflection is \( x = -3 \):**
\[
R' = (2(-3) - (-11), 1) = (-6 + 11, 1) = (5, 1)
\]
For each specific vertical line of reflection, the resulting coordinates of \( R' \) will vary.
### Conclusion
To provide a precise answer for \( R' \)'s coordinates, please confirm the vertical line of reflection that triangle \( PQR \) is being reflected over. If the line was the y-axis (\( x = 0 \)), then \( R' = (11, 1) \).
Answered by
GPT-4o mini
Thank you for providing the coordinates of triangle \( PQR \) and the coordinates of point \( P' \). Let's summarize the provided information:
- \( P(-8, 9) \)
- \( Q(-12, 7) \)
- \( R(-11, 1) \)
- \( P'(-6, 9) \)
Since the reflection of point \( P \) produces \( P' \), we can determine the line of reflection from these two points. The line of reflection will be the vertical line that is exactly halfway between points \( P \) and \( P' \).
### Finding the Line of Reflection:
The x-coordinates of \( P \) and \( P' \) are -8 and -6 respectively. To find the midpoint, we use the average of these x-coordinates:
\[
\text{Midpoint } x = \frac{-8 + (-6)}{2} = \frac{-14}{2} = -7
\]
Thus, the vertical line of reflection is \( x = -7 \).
### Reflecting Point \( R(-11, 1) \):
Now, we can reflect point \( R \) across the vertical line \( x = -7 \).
1. **Determine the distance from \( R \) to the line of reflection:**
- The x-coordinate of \( R \) is -11.
- The distance to the line \( x = -7 \) is:
\[
|-11 - (-7)| = |-11 + 7| = |-4| = 4
\]
2. **Since we want the reflected point \( R' \), we will move this distance in the opposite direction from the line of reflection:**
- The x-coordinate of \( R' \) will be:
\[
-7 + 4 = -3
\]
- The y-coordinate remains the same, as reflection over a vertical line does not change the y-coordinate:
\[
y_R' = y_R = 1
\]
So, the coordinates of point \( R' \) are:
\[
R' = (-3, 1)
\]
### Final Answer:
The coordinates of point \( R' \) after reflection over the vertical line \( x = -7 \) are \( (-3, 1) \).
- \( P(-8, 9) \)
- \( Q(-12, 7) \)
- \( R(-11, 1) \)
- \( P'(-6, 9) \)
Since the reflection of point \( P \) produces \( P' \), we can determine the line of reflection from these two points. The line of reflection will be the vertical line that is exactly halfway between points \( P \) and \( P' \).
### Finding the Line of Reflection:
The x-coordinates of \( P \) and \( P' \) are -8 and -6 respectively. To find the midpoint, we use the average of these x-coordinates:
\[
\text{Midpoint } x = \frac{-8 + (-6)}{2} = \frac{-14}{2} = -7
\]
Thus, the vertical line of reflection is \( x = -7 \).
### Reflecting Point \( R(-11, 1) \):
Now, we can reflect point \( R \) across the vertical line \( x = -7 \).
1. **Determine the distance from \( R \) to the line of reflection:**
- The x-coordinate of \( R \) is -11.
- The distance to the line \( x = -7 \) is:
\[
|-11 - (-7)| = |-11 + 7| = |-4| = 4
\]
2. **Since we want the reflected point \( R' \), we will move this distance in the opposite direction from the line of reflection:**
- The x-coordinate of \( R' \) will be:
\[
-7 + 4 = -3
\]
- The y-coordinate remains the same, as reflection over a vertical line does not change the y-coordinate:
\[
y_R' = y_R = 1
\]
So, the coordinates of point \( R' \) are:
\[
R' = (-3, 1)
\]
### Final Answer:
The coordinates of point \( R' \) after reflection over the vertical line \( x = -7 \) are \( (-3, 1) \).
Answered by
GPT-4o mini
Given the vertices of triangle \( PQR \) with the following coordinates:
- \( P(-8, 9) \)
- \( Q(-12, 7) \)
- \( R(-11, 1) \)
Based on the previous messages, you mentioned reflecting point \( P \) to obtain point \( P'(-6, 9) \). This suggests that the reflection is taking place across a vertical line.
### Finding the Line of Reflection
To identify the line of reflection, we can find the midpoint of the line segment connecting \( P(-8, 9) \) and \( P'(-6, 9) \):
1. **Calculate the midpoint x-coordinate:**
\[
\text{Midpoint } x = \frac{x_P + x_{P'}}{2} = \frac{-8 + (-6)}{2} = \frac{-14}{2} = -7
\]
This indicates that the vertical line of reflection is \( x = -7 \).
### Reflecting Point R
Now, let’s reflect point \( R \) across this vertical line \( x = -7 \):
1. **Distance from \( R \) to the line of reflection:**
- The x-coordinate of \( R \) is -11.
- The distance to the line \( x = -7 \):
\[
|-11 - (-7)| = |-11 + 7| = |-4| = 4
\]
2. **To find \( R' \), we move 4 units in the opposite direction from the line:**
- The x-coordinate of \( R' \) becomes:
\[
-7 + 4 = -3
\]
- The y-coordinate remains unchanged, so:
\[
y_{R'} = y_R = 1
\]
Thus, the coordinates of point \( R' \) after reflecting \( R \) across the line of reflection \( x = -7 \) are:
\[
R' = (-3, 1)
\]
### Summary
- \( R' \) after the reflection is at the coordinates \( (-3, 1) \).
- \( P(-8, 9) \)
- \( Q(-12, 7) \)
- \( R(-11, 1) \)
Based on the previous messages, you mentioned reflecting point \( P \) to obtain point \( P'(-6, 9) \). This suggests that the reflection is taking place across a vertical line.
### Finding the Line of Reflection
To identify the line of reflection, we can find the midpoint of the line segment connecting \( P(-8, 9) \) and \( P'(-6, 9) \):
1. **Calculate the midpoint x-coordinate:**
\[
\text{Midpoint } x = \frac{x_P + x_{P'}}{2} = \frac{-8 + (-6)}{2} = \frac{-14}{2} = -7
\]
This indicates that the vertical line of reflection is \( x = -7 \).
### Reflecting Point R
Now, let’s reflect point \( R \) across this vertical line \( x = -7 \):
1. **Distance from \( R \) to the line of reflection:**
- The x-coordinate of \( R \) is -11.
- The distance to the line \( x = -7 \):
\[
|-11 - (-7)| = |-11 + 7| = |-4| = 4
\]
2. **To find \( R' \), we move 4 units in the opposite direction from the line:**
- The x-coordinate of \( R' \) becomes:
\[
-7 + 4 = -3
\]
- The y-coordinate remains unchanged, so:
\[
y_{R'} = y_R = 1
\]
Thus, the coordinates of point \( R' \) after reflecting \( R \) across the line of reflection \( x = -7 \) are:
\[
R' = (-3, 1)
\]
### Summary
- \( R' \) after the reflection is at the coordinates \( (-3, 1) \).