Printable math worksheet for practicing geometric transformations of triangles on a coordinate plane.

Problem Description:

The task involves transforming triangles on a coordinate grid according to specific geometric transformations. Each transformation is described, and the goal is to graph the resulting triangle after applying the transformation.

Solution Explanation:

#### 1. Translate 2 units up and 7 units right
- Transformation: Move each vertex of the triangle 2 units up and 7 units to the right.
- Steps:
1. Identify the coordinates of the vertices of the original triangle.
2. Add 2 to the y-coordinate and 7 to the x-coordinate of each vertex.
3. Plot the new vertices and connect them to form the transformed triangle.

#### 2. Reflection across the line \( x = 3 \)
- Transformation: Reflect each vertex of the triangle across the vertical line \( x = 3 \).
- Steps:
1. Identify the coordinates of the vertices of the original triangle.
2. For each vertex \((x, y)\), calculate the reflected point using the formula:
\[
(x', y') = (6 - x, y)
\]
where \( x' \) is the new x-coordinate and \( y' \) remains the same.
3. Plot the new vertices and connect them to form the transformed triangle.

#### 3. 90° clockwise rotation about the origin
- Transformation: Rotate each vertex of the triangle 90° clockwise around the origin.
- Steps:
1. Identify the coordinates of the vertices of the original triangle.
2. For each vertex \((x, y)\), calculate the rotated point using the formula:
\[
(x', y') = (y, -x)
\]
3. Plot the new vertices and connect them to form the transformed triangle.

#### 4. Translate 4 units left and 6 units down
- Transformation: Move each vertex of the triangle 4 units to the left and 6 units down.
- Steps:
1. Identify the coordinates of the vertices of the original triangle.
2. Subtract 4 from the x-coordinate and 6 from the y-coordinate of each vertex.
3. Plot the new vertices and connect them to form the transformed triangle.

#### 5. Reflection across the line \( y = -1 \)
- Transformation: Reflect each vertex of the triangle across the horizontal line \( y = -1 \).
- Steps:
1. Identify the coordinates of the vertices of the original triangle.
2. For each vertex \((x, y)\), calculate the reflected point using the formula:
\[
(x', y') = (x, -2 - y)
\]
where \( x' \) remains the same and \( y' \) is calculated based on the reflection.
3. Plot the new vertices and connect them to form the transformed triangle.

#### 6. 90° counterclockwise rotation about the origin
- Transformation: Rotate each vertex of the triangle 90° counterclockwise around the origin.
- Steps:
1. Identify the coordinates of the vertices of the original triangle.
2. For each vertex \((x, y)\), calculate the rotated point using the formula:
\[
(x', y') = (-y, x)
\]
3. Plot the new vertices and connect them to form the transformed triangle.

#### 7. Translate 5 units down and 1 unit right
- Transformation: Move each vertex of the triangle 5 units down and 1 unit to the right.
- Steps:
1. Identify the coordinates of the vertices of the original triangle.
2. Add 1 to the x-coordinate and subtract 5 from the y-coordinate of each vertex.
3. Plot the new vertices and connect them to form the transformed triangle.

#### 8. Reflection across the line \( y = x \)
- Transformation: Reflect each vertex of the triangle across the line \( y = x \).
- Steps:
1. Identify the coordinates of the vertices of the original triangle.
2. For each vertex \((x, y)\), calculate the reflected point by swapping the x and y coordinates:
\[
(x', y') = (y, x)
\]
3. Plot the new vertices and connect them to form the transformed triangle.

Final Answer:

Each transformation requires identifying the original vertices, applying the transformation rules, and plotting the new vertices. The final answer for each transformation is the graph of the transformed triangle.

\[
\boxed{\text{Graph each transformed triangle as described above.}}
\]