Reflection, Translation, and Rotation Worksheets - Free Printable

Problem Overview:

The task involves reflecting triangles across the x-axis and recording the new coordinates of the reflected points. The goal is to understand how reflection across the x-axis affects the coordinates of points.

Key Concept:

- Reflection across the x-axis: When a point \((x, y)\) is reflected across the x-axis, its new coordinates become \((x, -y)\). The x-coordinate remains the same, but the y-coordinate changes sign.

Solution:

#### Step 1: Reflect Triangle ABC Across the x-axis
1. Identify the original coordinates of Triangle ABC:
- From the grid in the image:
- \( A = (2, 4) \)
- \( B = (5, 4) \)
- \( C = (3, 6) \)

2. Apply the reflection rule \((x, y) \rightarrow (x, -y)\):
- For \( A(2, 4) \):
\[
A' = (2, -4)
\]
- For \( B(5, 4) \):
\[
B' = (5, -4)
\]
- For \( C(3, 6) \):
\[
C' = (3, -6)
\]

3. Record the new coordinates:
- \( A' = (2, -4) \)
- \( B' = (5, -4) \)
- \( C' = (3, -6) \)

4. Plot the reflected triangle \( A'B'C' \) on the grid.

#### Step 2: Reflect Triangle EFG Across the x-axis
1. Identify the original coordinates of Triangle EFG:
- From the grid in the image:
- \( E = (1, 3) \)
- \( F = (4, 3) \)
- \( G = (3, 5) \)

2. Apply the reflection rule \((x, y) \rightarrow (x, -y)\):
- For \( E(1, 3) \):
\[
E' = (1, -3)
\]
- For \( F(4, 3) \):
\[
F' = (4, -3)
\]
- For \( G(3, 5) \):
\[
G' = (3, -5)
\]

3. Record the new coordinates:
- \( E' = (1, -3) \)
- \( F' = (4, -3) \)
- \( G' = (3, -5) \)

4. Plot the reflected triangle \( E'F'G' \) on the grid.

#### Step 3: Reflect Triangle XYZ Across the x-axis
1. Identify the original coordinates of Triangle XYZ:
- From the grid in the image:
- \( X = (1, 2) \)
- \( Y = (4, 2) \)
- \( Z = (3, 4) \)

2. Apply the reflection rule \((x, y) \rightarrow (x, -y)\):
- For \( X(1, 2) \):
\[
X' = (1, -2)
\]
- For \( Y(4, 2) \):
\[
Y' = (4, -2)
\]
- For \( Z(3, 4) \):
\[
Z' = (3, -4)
\]

3. Record the new coordinates:
- \( X' = (1, -2) \)
- \( Y' = (4, -2) \)
- \( Z' = (3, -4) \)

4. Plot the reflected triangle \( X'Y'Z' \) on the grid.

Final Answer:

\[
\boxed{
\begin{array}{ll}
\text{1. Triangle ABC:} & \\
A' = (2, -4) & B' = (5, -4) & C' = (3, -6) \\
\text{2. Triangle EFG:} & \\
E' = (1, -3) & F' = (4, -3) & G' = (3, -5) \\
\text{3. Triangle XYZ:} & \\
X' = (1, -2) & Y' = (4, -2) & Z' = (3, -4) \\
\end{array}
}
\]

Explanation:

- Each point's x-coordinate remains unchanged.
- The y-coordinate of each point is negated (multiplied by -1) to reflect it across the x-axis.
- The reflected triangles are plotted by connecting the new points.