Question

9.1 ... figure abc. draw the resulting figure after each transformation. a) translate figure abc 5 units to the right and 2 units up. b) rotate figure abc 180° clockwise around point a. c) reflect figure abc across the line k.

Explanation:

Part (a): Translation of Figure \( ABC \)

Step 1: Identify Coordinates of Vertices

First, determine the coordinates of points \( A \), \( B \), and \( C \) from the grid. Let's assume the grid has a standard coordinate system where each square is 1 unit. Suppose:

• \( A = (x_A, y_A) \)
• \( B = (x_B, y_B) \)
• \( C = (x_C, y_C) \)

Step 2: Apply Translation

To translate 5 units right and 2 units up, use the translation rule \((x, y) \to (x + 5, y + 2)\) for each vertex:

• New \( A' = (x_A + 5, y_A + 2) \)
• New \( B' = (x_B + 5, y_B + 2) \)
• New \( C' = (x_C + 5, y_C + 2) \)

Step 3: Draw the Translated Figure

Plot the new points \( A' \), \( B' \), and \( C' \) on the grid and connect them to form the translated triangle \( A'B'C' \).

Part (b): Rotation of Figure \( ABC \) 180° Counterclockwise about Point \( A \)

Step 1: Recall Rotation Rule

The rule for a 180° counterclockwise rotation about a point \( (a, b) \) is \((x, y) \to (2a - x, 2b - y)\). Let \( A = (a, b) \), \( B = (x_B, y_B) \), \( C = (x_C, y_C) \).

Step 2: Rotate Points \( B \) and \( C \)

• For point \( B \): New \( B' = (2a - x_B, 2b - y_B) \)
• For point \( C \): New \( C' = (2a - x_C, 2b - y_C) \) (Point \( A \) remains \( A \) since it's the center of rotation)

Step 3: Draw the Rotated Figure

Plot \( A \), \( B' \), and \( C' \) on the grid and connect them to form the rotated triangle \( AB'C' \).

Part (c): Reflection of Figure \( ABC \) across Line \( K \)

Step 1: Determine the Line of Reflection

Line \( K \) is a vertical (or horizontal) line. For a vertical line \( x = k \), the reflection rule is \((x, y) \to (2k - x, y)\). For a horizontal line \( y = k \), the rule is \((x, y) \to (x, 2k - y)\).

Step 2: Reflect Each Vertex

• Let \( A = (x_A, y_A) \), \( B = (x_B, y_B) \), \( C = (x_C, y_C) \)
• Apply the reflection rule to each point to get \( A' \), \( B' \), \( C' \)

Step 3: Draw the Reflected Figure

Plot the reflected points \( A' \), \( B' \), \( C' \) on the grid and connect them to form the reflected triangle \( A'B'C' \).

Final Answer (Visual Instructions)

• (a) Translate each vertex 5 right, 2 up, redraw.
• (b) Rotate \( B, C \) 180° about \( A \), redraw.
• (c) Reflect each vertex over line \( K \), redraw.

(Note: Actual drawing requires plotting on the grid using the coordinate transformations above.)

Answer:

Part (a): Translation of Figure \( ABC \)

Step 1: Identify Coordinates of Vertices

First, determine the coordinates of points \( A \), \( B \), and \( C \) from the grid. Let's assume the grid has a standard coordinate system where each square is 1 unit. Suppose:

• \( A = (x_A, y_A) \)
• \( B = (x_B, y_B) \)
• \( C = (x_C, y_C) \)

Step 2: Apply Translation

To translate 5 units right and 2 units up, use the translation rule \((x, y) \to (x + 5, y + 2)\) for each vertex:

• New \( A' = (x_A + 5, y_A + 2) \)
• New \( B' = (x_B + 5, y_B + 2) \)
• New \( C' = (x_C + 5, y_C + 2) \)

Step 3: Draw the Translated Figure

Plot the new points \( A' \), \( B' \), and \( C' \) on the grid and connect them to form the translated triangle \( A'B'C' \).

Part (b): Rotation of Figure \( ABC \) 180° Counterclockwise about Point \( A \)

Step 1: Recall Rotation Rule

The rule for a 180° counterclockwise rotation about a point \( (a, b) \) is \((x, y) \to (2a - x, 2b - y)\). Let \( A = (a, b) \), \( B = (x_B, y_B) \), \( C = (x_C, y_C) \).

Step 2: Rotate Points \( B \) and \( C \)

• For point \( B \): New \( B' = (2a - x_B, 2b - y_B) \)
• For point \( C \): New \( C' = (2a - x_C, 2b - y_C) \) (Point \( A \) remains \( A \) since it's the center of rotation)

Step 3: Draw the Rotated Figure

Plot \( A \), \( B' \), and \( C' \) on the grid and connect them to form the rotated triangle \( AB'C' \).

Part (c): Reflection of Figure \( ABC \) across Line \( K \)

Step 1: Determine the Line of Reflection

Line \( K \) is a vertical (or horizontal) line. For a vertical line \( x = k \), the reflection rule is \((x, y) \to (2k - x, y)\). For a horizontal line \( y = k \), the rule is \((x, y) \to (x, 2k - y)\).

Step 2: Reflect Each Vertex

• Let \( A = (x_A, y_A) \), \( B = (x_B, y_B) \), \( C = (x_C, y_C) \)
• Apply the reflection rule to each point to get \( A' \), \( B' \), \( C' \)

Step 3: Draw the Reflected Figure

Plot the reflected points \( A' \), \( B' \), \( C' \) on the grid and connect them to form the reflected triangle \( A'B'C' \).

Final Answer (Visual Instructions)

• (a) Translate each vertex 5 right, 2 up, redraw.
• (b) Rotate \( B, C \) 180° about \( A \), redraw.
• (c) Reflect each vertex over line \( K \), redraw.

(Note: Actual drawing requires plotting on the grid using the coordinate transformations above.)