SOLUTION: Rotations of shapes - Studypool - Free Printable

Problem Analysis:

The task involves performing various transformations (reflections) on geometric figures using a coordinate plane. The goal is to reflect the given shapes across specific lines (e.g., the x-axis, y-axis, or other specified lines) and plot the resulting images.

Solution Explanation:

#### Part I: Reflections

1. Reflection of Triangle ABC Across the x-axis:
- Original Coordinates: Assume the vertices of triangle \( \triangle ABC \) are \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \).
- Reflection Rule Across the x-axis: When reflecting a point \((x, y)\) across the x-axis, the new coordinates become \((x, -y)\).
- New Coordinates:
- \( A'(x_1, -y_1) \)
- \( B'(x_2, -y_2) \)
- \( C'(x_3, -y_3) \)
- Graphing: Plot the new points \( A' \), \( B' \), and \( C' \) and connect them to form the reflected triangle.

2. Reflection of Triangle DEF Across the Line \( y = x \):
- Original Coordinates: Assume the vertices of triangle \( \triangle DEF \) are \( D(x_1, y_1) \), \( E(x_2, y_2) \), and \( F(x_3, y_3) \).
- Reflection Rule Across the Line \( y = x \): When reflecting a point \((x, y)\) across the line \( y = x \), the new coordinates become \((y, x)\).
- New Coordinates:
- \( D'(y_1, x_1) \)
- \( E'(y_2, x_2) \)
- \( F'(y_3, x_3) \)
- Graphing: Plot the new points \( D' \), \( E' \), and \( F' \) and connect them to form the reflected triangle.

3. Reflection of Quadrilateral WXYZ Across the y-axis:
- Original Coordinates: Assume the vertices of quadrilateral \( WXYZ \) are \( W(x_1, y_1) \), \( X(x_2, y_2) \), \( Y(x_3, y_3) \), and \( Z(x_4, y_4) \).
- Reflection Rule Across the y-axis: When reflecting a point \((x, y)\) across the y-axis, the new coordinates become \((-x, y)\).
- New Coordinates:
- \( W'(-x_1, y_1) \)
- \( X'(-x_2, y_2) \)
- \( Y'(-x_3, y_3) \)
- \( Z'(-x_4, y_4) \)
- Graphing: Plot the new points \( W' \), \( X' \), \( Y' \), and \( Z' \) and connect them to form the reflected quadrilateral.

4. Reflection of Triangle LMN Across the x-axis:
- Original Coordinates: Assume the vertices of triangle \( \triangle LMN \) are \( L(x_1, y_1) \), \( M(x_2, y_2) \), and \( N(x_3, y_3) \).
- Reflection Rule Across the x-axis: When reflecting a point \((x, y)\) across the x-axis, the new coordinates become \((x, -y)\).
- New Coordinates:
- \( L'(x_1, -y_1) \)
- \( M'(x_2, -y_2) \)
- \( N'(x_3, -y_3) \)
- Graphing: Plot the new points \( L' \), \( M' \), and \( N' \) and connect them to form the reflected triangle.

5. Reflection of Quadrilateral PQRS Across the Line \( y = -x \):
- Original Coordinates: Assume the vertices of quadrilateral \( PQRS \) are \( P(x_1, y_1) \), \( Q(x_2, y_2) \), \( R(x_3, y_3) \), and \( S(x_4, y_4) \).
- Reflection Rule Across the Line \( y = -x \): When reflecting a point \((x, y)\) across the line \( y = -x \), the new coordinates become \((-y, -x)\).
- New Coordinates:
- \( P'(-y_1, -x_1) \)
- \( Q'(-y_2, -x_2) \)
- \( R'(-y_3, -x_3) \)
- \( S'(-y_4, -x_4) \)
- Graphing: Plot the new points \( P' \), \( Q' \), \( R' \), and \( S' \) and connect them to form the reflected quadrilateral.

6. Reflection of Triangle GHI Across the Line \( y = 2 \):
- Original Coordinates: Assume the vertices of triangle \( \triangle GHI \) are \( G(x_1, y_1) \), \( H(x_2, y_2) \), and \( I(x_3, y_3) \).
- Reflection Rule Across the Line \( y = 2 \): To reflect a point \((x, y)\) across the line \( y = 2 \), the new y-coordinate is calculated as \( y' = 2 \times 2 - y = 4 - y \). The x-coordinate remains the same.
- New Coordinates:
- \( G'(x_1, 4 - y_1) \)
- \( H'(x_2, 4 - y_2) \)
- \( I'(x_3, 4 - y_3) \)
- Graphing: Plot the new points \( G' \), \( H' \), and \( I' \) and connect them to form the reflected triangle.

Final Answer:

\[
\boxed{
\text{See detailed steps above for each reflection.}
}
\]