Question
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Part 1
Given triangle symbol JKL with vertices Upper Jleft parenthesis negative 4 comma negative 4 right parenthesis, Upper Kleft parenthesis negative 5 comma negative 5 right parenthesis, and Upper Lleft parenthesis negative 3 comma negative 1 right parenthesis, write the translation equivalent to the composition of transformations. The equation of line t is xequals8.
Upper R Subscript t Baseline circle Upper R Subscript y dash axis
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Part 1
left parenthesis Upper R Subscript t Baseline circle Upper R Subscript y dash axis right parenthesis left parenthesis triangle symbol JKL right parenthesisequalsUpper T Subscript left angle nothing comma nothing right angle Baseline left parenthesis triangle symbol JKL right parenthesis
Part 1
Given triangle symbol JKL with vertices Upper Jleft parenthesis negative 4 comma negative 4 right parenthesis, Upper Kleft parenthesis negative 5 comma negative 5 right parenthesis, and Upper Lleft parenthesis negative 3 comma negative 1 right parenthesis, write the translation equivalent to the composition of transformations. The equation of line t is xequals8.
Upper R Subscript t Baseline circle Upper R Subscript y dash axis
Question content area bottom
Part 1
left parenthesis Upper R Subscript t Baseline circle Upper R Subscript y dash axis right parenthesis left parenthesis triangle symbol JKL right parenthesisequalsUpper T Subscript left angle nothing comma nothing right angle Baseline left parenthesis triangle symbol JKL right parenthesis
Answers
Answered by
GPT-4o mini
To find the resulting transformation from the composition of transformations given in the problem, we need to analyze each part of the transformation sequence.
1. **Transformation \( R_t \)**:
- The notation \( R_t \) typically refers to a reflection across a line. In this case, the line is given as \( x = 8 \).
- Reflecting a point \( (x, y) \) over the line \( x = a \) leads to the transformation formula:
\[
(x', y) = (2a - x, y)
\]
- For our case, reflecting over \( x = 8 \) results in:
\[
(x', y) = (2 \cdot 8 - x, y) = (16 - x, y)
\]
2. **Transformation \( R_{y' \text{-axis}} \)**:
- This notation refers to a reflection across the \( y' \)-axis (which is often interpreted as the y-axis).
- Reflecting a point \( (x, y) \) over the y-axis results in the transformation:
\[
(x', y') = (-x, y)
\]
3. **Applying the transformations to triangle \( JKL \)**:
- We will apply the transformations to each of the vertices \( J(-4, -4) \), \( K(-5, -5) \), and \( L(-3, -1) \).
**Step 1: Reflection over the line \( x = 8 \)**:
- For \( J(-4, -4) \):
\[
J' = (16 - (-4), -4) = (20, -4)
\]
- For \( K(-5, -5) \):
\[
K' = (16 - (-5), -5) = (21, -5)
\]
- For \( L(-3, -1) \):
\[
L' = (16 - (-3), -1) = (19, -1)
\]
**Step 2: Reflection over the y-axis**:
- Now reflecting the new points \( J' \), \( K' \), and \( L' \):
- For \( J'(20, -4) \):
\[
J'' = (-20, -4)
\]
- For \( K'(21, -5) \):
\[
K'' = (-21, -5)
\]
- For \( L'(19, -1) \):
\[
L'' = (-19, -1)
\]
4. **Final Transformed Triangle**:
- The transformed triangle vertices are \( J''(-20, -4) \), \( K''(-21, -5) \), and \( L''(-19, -1) \).
5. **Result**:
- The original triangle \( JKL \) is transformed to triangle \( J''K''L'' \) after the composition of transformations \( R_t \circ R_{y' \text{-axis}} \).
6. **Description of the Transformation**:
- The overall transformation can be described as a **translation** defined by taking each point \( (x, y) \) in triangle \( JKL \) and mapping it to the final points \( (-20, -4) \), \( (-21, -5) \), and \( (-19, -1) \).
- The specific translation vector can be calculated:
- For example, the movement of point \( J(-4, -4) \) to \( J''(-20, -4) \) can be described as moving left \( 16 \) units (from \( -4 \) to \( -20 \)).
Thus, the composition of transformations can be succinctly summarized:
\[
\text{Triangle } JKL \text{ is transformed to } T(-20, -4), T(-21, -5), T(-19, -1) \text{ through a series of reflections.}
\]
This final result illustrates that the shape has been moved substantially in the plane through the specified sequence of transformations.
1. **Transformation \( R_t \)**:
- The notation \( R_t \) typically refers to a reflection across a line. In this case, the line is given as \( x = 8 \).
- Reflecting a point \( (x, y) \) over the line \( x = a \) leads to the transformation formula:
\[
(x', y) = (2a - x, y)
\]
- For our case, reflecting over \( x = 8 \) results in:
\[
(x', y) = (2 \cdot 8 - x, y) = (16 - x, y)
\]
2. **Transformation \( R_{y' \text{-axis}} \)**:
- This notation refers to a reflection across the \( y' \)-axis (which is often interpreted as the y-axis).
- Reflecting a point \( (x, y) \) over the y-axis results in the transformation:
\[
(x', y') = (-x, y)
\]
3. **Applying the transformations to triangle \( JKL \)**:
- We will apply the transformations to each of the vertices \( J(-4, -4) \), \( K(-5, -5) \), and \( L(-3, -1) \).
**Step 1: Reflection over the line \( x = 8 \)**:
- For \( J(-4, -4) \):
\[
J' = (16 - (-4), -4) = (20, -4)
\]
- For \( K(-5, -5) \):
\[
K' = (16 - (-5), -5) = (21, -5)
\]
- For \( L(-3, -1) \):
\[
L' = (16 - (-3), -1) = (19, -1)
\]
**Step 2: Reflection over the y-axis**:
- Now reflecting the new points \( J' \), \( K' \), and \( L' \):
- For \( J'(20, -4) \):
\[
J'' = (-20, -4)
\]
- For \( K'(21, -5) \):
\[
K'' = (-21, -5)
\]
- For \( L'(19, -1) \):
\[
L'' = (-19, -1)
\]
4. **Final Transformed Triangle**:
- The transformed triangle vertices are \( J''(-20, -4) \), \( K''(-21, -5) \), and \( L''(-19, -1) \).
5. **Result**:
- The original triangle \( JKL \) is transformed to triangle \( J''K''L'' \) after the composition of transformations \( R_t \circ R_{y' \text{-axis}} \).
6. **Description of the Transformation**:
- The overall transformation can be described as a **translation** defined by taking each point \( (x, y) \) in triangle \( JKL \) and mapping it to the final points \( (-20, -4) \), \( (-21, -5) \), and \( (-19, -1) \).
- The specific translation vector can be calculated:
- For example, the movement of point \( J(-4, -4) \) to \( J''(-20, -4) \) can be described as moving left \( 16 \) units (from \( -4 \) to \( -20 \)).
Thus, the composition of transformations can be succinctly summarized:
\[
\text{Triangle } JKL \text{ is transformed to } T(-20, -4), T(-21, -5), T(-19, -1) \text{ through a series of reflections.}
\]
This final result illustrates that the shape has been moved substantially in the plane through the specified sequence of transformations.