Question
Suppose △ABC is graphed on a coordinate plane along with two lines, p which has equation x = 2, and q which has equation x = –1. What translation is equivalent to (Rp ∘ Rq) (ΔABC)?
(Rp ∘ Rq)(ΔABC) = T<m, n>(ΔABC) where m =
and n =
.
(Rp ∘ Rq)(ΔABC) = T<m, n>(ΔABC) where m =
and n =
.
Answers
GPT-4o mini
To find the translation equivalent to the composition of reflections \( (R_p \circ R_q)(\Delta ABC) \), where line \( p \) has the equation \( x = 2 \) and line \( q \) has the equation \( x = -1 \), we will first analyze the effects of each reflection on the points in triangle \( \Delta ABC \).
1. **Reflection over line \( p \) (x = 2)**:
- When reflecting a point \( (x, y) \) across the line \( x = 2 \), the x-coordinate is transformed as follows:
\[
x' = 2 + (2 - x) = 4 - x
\]
- Thus, the reflection \( R_p \) will change \( (x, y) \) to \( (4 - x, y) \).
2. **Reflection over line \( q \) (x = -1)**:
- Reflecting a point \( (x, y) \) over the line \( x = -1 \) modifies the x-coordinate as follows:
\[
x'' = -1 + (-1 - x) = -2 - x
\]
- This means that the reflection \( R_q \) will change \( (x', y) \) to \( (-2 - x', y) \).
Now, we apply these transformations sequentially:
1. Start with \( (x, y) \):
After reflection over \( q \):
\[
(x', y) = R_q(x, y) = (-2 - (4 - x), y) = (-6 + x, y)
\]
2. Then reflect this result over \( p \):
\[
(x'', y) = R_p(x', y) = (4 - (-6 + x), y) = (4 + 6 - x, y) = (10 - x, y)
\]
After both reflections, the new coordinates of each point of triangle \( \Delta ABC \) can be expressed as:
\[
(x'', y) = (10 - x, y)
\]
From this, we can deduce that the final transformation can be described as a translation \( T<m, n> \) where:
- The x-coordinate is moved from \( x \) to \( 10 - x \), which indicates a translation of \( m = 10 - 2x \), but observing that the effective change is \( (10 - 2) = 8 \) when x equals \( 2 \).
- The y-coordinate remains unchanged (i.e., \( n = 0 \)).
So the translation can be represented as:
\[
T<m, n> = T<8, 0>
\]
Consequently, the values are:
- \( m = 8 \)
- \( n = 0 \)
Thus the translation equivalent to \( (R_p \circ R_q)(\Delta ABC) \) is \( T<8, 0> \), where the final answers are:
- \( m = 8 \)
- \( n = 0 \)
1. **Reflection over line \( p \) (x = 2)**:
- When reflecting a point \( (x, y) \) across the line \( x = 2 \), the x-coordinate is transformed as follows:
\[
x' = 2 + (2 - x) = 4 - x
\]
- Thus, the reflection \( R_p \) will change \( (x, y) \) to \( (4 - x, y) \).
2. **Reflection over line \( q \) (x = -1)**:
- Reflecting a point \( (x, y) \) over the line \( x = -1 \) modifies the x-coordinate as follows:
\[
x'' = -1 + (-1 - x) = -2 - x
\]
- This means that the reflection \( R_q \) will change \( (x', y) \) to \( (-2 - x', y) \).
Now, we apply these transformations sequentially:
1. Start with \( (x, y) \):
After reflection over \( q \):
\[
(x', y) = R_q(x, y) = (-2 - (4 - x), y) = (-6 + x, y)
\]
2. Then reflect this result over \( p \):
\[
(x'', y) = R_p(x', y) = (4 - (-6 + x), y) = (4 + 6 - x, y) = (10 - x, y)
\]
After both reflections, the new coordinates of each point of triangle \( \Delta ABC \) can be expressed as:
\[
(x'', y) = (10 - x, y)
\]
From this, we can deduce that the final transformation can be described as a translation \( T<m, n> \) where:
- The x-coordinate is moved from \( x \) to \( 10 - x \), which indicates a translation of \( m = 10 - 2x \), but observing that the effective change is \( (10 - 2) = 8 \) when x equals \( 2 \).
- The y-coordinate remains unchanged (i.e., \( n = 0 \)).
So the translation can be represented as:
\[
T<m, n> = T<8, 0>
\]
Consequently, the values are:
- \( m = 8 \)
- \( n = 0 \)
Thus the translation equivalent to \( (R_p \circ R_q)(\Delta ABC) \) is \( T<8, 0> \), where the final answers are:
- \( m = 8 \)
- \( n = 0 \)