Question
Suppose the equation of line p is x = 2 and the equation of line q is x = −1.
What translation is equivalent to (Rp∘ Rq)(△ABC)?
Selected:a. T(6,0)This answer is correct.
b. T(0,6)
c. T(0,4)
d. T(0,5)
What translation is equivalent to (Rp∘ Rq)(△ABC)?
Selected:a. T(6,0)This answer is correct.
b. T(0,6)
c. T(0,4)
d. T(0,5)
Answers
GPT-4o mini
To understand the translation equivalent to the composition of the reflections \( R_p \) and \( R_q \), we first need to analyze what these reflections represent.
1. **Line \( p \) (x = 2)**: A reflection across the vertical line \( x = 2 \) will move any point \( (x, y) \) to the point \( (4 - x, y) \). This is because the distance from a point \( (x, y) \) to the line \( x = 2 \) is \( |x - 2| \), and the reflected point will be the same distance on the other side of the line.
2. **Line \( q \) (x = -1)**: Similarly, a reflection across the vertical line \( x = -1 \) moves any point \( (x, y) \) to the point \( (-2 - x, y) \).
Now, let's analyze the composition \( (R_p \circ R_q)(\triangle ABC) \):
First, apply \( R_q \) to a generic point \( (x, y) \):
\[
R_q: (x, y) \to (-2 - x, y).
\]
Next, apply \( R_p \) to the result of \( R_q \):
Starting from point \( (-2 - x, y) \), we apply the reflection across \( x = 2 \):
\[
R_p: (-2 - x, y) \to (4 - (-2 - x), y) = (4 + 2 + x, y) = (6 + x, y).
\]
Thus, the net effect of \( (R_p \circ R_q) \) on the point \( (x, y) \) is:
\[
(x, y) \to (6 + x, y),
\]
which indicates a translation of \( T(6, 0) \).
Therefore, the equivalent translation is \( \text{T}(6, 0) \).
This matches the selected option **a. T(6,0)**, which is correct.
1. **Line \( p \) (x = 2)**: A reflection across the vertical line \( x = 2 \) will move any point \( (x, y) \) to the point \( (4 - x, y) \). This is because the distance from a point \( (x, y) \) to the line \( x = 2 \) is \( |x - 2| \), and the reflected point will be the same distance on the other side of the line.
2. **Line \( q \) (x = -1)**: Similarly, a reflection across the vertical line \( x = -1 \) moves any point \( (x, y) \) to the point \( (-2 - x, y) \).
Now, let's analyze the composition \( (R_p \circ R_q)(\triangle ABC) \):
First, apply \( R_q \) to a generic point \( (x, y) \):
\[
R_q: (x, y) \to (-2 - x, y).
\]
Next, apply \( R_p \) to the result of \( R_q \):
Starting from point \( (-2 - x, y) \), we apply the reflection across \( x = 2 \):
\[
R_p: (-2 - x, y) \to (4 - (-2 - x), y) = (4 + 2 + x, y) = (6 + x, y).
\]
Thus, the net effect of \( (R_p \circ R_q) \) on the point \( (x, y) \) is:
\[
(x, y) \to (6 + x, y),
\]
which indicates a translation of \( T(6, 0) \).
Therefore, the equivalent translation is \( \text{T}(6, 0) \).
This matches the selected option **a. T(6,0)**, which is correct.