Question

1. rotation 90° counterclockwise about the origin

Explanation:

To solve a \( 90^\circ \) counterclockwise rotation about the origin, we use the transformation rule: for a point \( (x, y) \), the image after a \( 90^\circ \) counterclockwise rotation about the origin is \( (-y, x) \).

Step 1: Identify Coordinates of Original Points

First, we determine the coordinates of the vertices of the original figure (let's assume the figure has vertices, say, \( P, E, W, H \) from the grid). For example, if we identify:

• \( P(-3, -2) \)
• \( E(-2, -3) \)
• \( W(-1, 1) \)
• \( H(1, 0) \)

Step 2: Apply the Rotation Rule

For each point \( (x, y) \), the new coordinates \( (x', y') \) after \( 90^\circ \) counterclockwise rotation are \( x' = -y \) and \( y' = x \).

• For \( P(-3, -2) \):

\( x' = -(-2) = 2 \), \( y' = -3 \) → \( P'(2, -3) \)

• For \( E(-2, -3) \):

\( x' = -(-3) = 3 \), \( y' = -2 \) → \( E'(3, -2) \)

• For \( W(-1, 1) \):

\( x' = -1 \), \( y' = -1 \) → \( W'(-1, -1) \)

• For \( H(1, 0) \):

\( x' = -0 = 0 \), \( y' = 1 \) → \( H'(0, 1) \)

Step 3: Plot the Rotated Points

Using the new coordinates \( P'(2, -3) \), \( E'(3, -2) \), \( W'(-1, -1) \), and \( H'(0, 1) \), plot these points on the grid and connect them to form the rotated figure.

Final Answer

The coordinates of the vertices after a \( 90^\circ \) counterclockwise rotation about the origin are:

• \( P(-3, -2)

ightarrow P'(2, -3) \)

• \( E(-2, -3)

ightarrow E'(3, -2) \)

• \( W(-1, 1)

ightarrow W'(-1, -1) \)

• \( H(1, 0)

ightarrow H'(0, 1) \)

(Plot these points to visualize the rotated figure.)

Answer:

To solve a \( 90^\circ \) counterclockwise rotation about the origin, we use the transformation rule: for a point \( (x, y) \), the image after a \( 90^\circ \) counterclockwise rotation about the origin is \( (-y, x) \).

Step 1: Identify Coordinates of Original Points

First, we determine the coordinates of the vertices of the original figure (let's assume the figure has vertices, say, \( P, E, W, H \) from the grid). For example, if we identify:

• \( P(-3, -2) \)
• \( E(-2, -3) \)
• \( W(-1, 1) \)
• \( H(1, 0) \)

Step 2: Apply the Rotation Rule

For each point \( (x, y) \), the new coordinates \( (x', y') \) after \( 90^\circ \) counterclockwise rotation are \( x' = -y \) and \( y' = x \).

• For \( P(-3, -2) \):

\( x' = -(-2) = 2 \), \( y' = -3 \) → \( P'(2, -3) \)

• For \( E(-2, -3) \):

\( x' = -(-3) = 3 \), \( y' = -2 \) → \( E'(3, -2) \)

• For \( W(-1, 1) \):

\( x' = -1 \), \( y' = -1 \) → \( W'(-1, -1) \)

• For \( H(1, 0) \):

\( x' = -0 = 0 \), \( y' = 1 \) → \( H'(0, 1) \)

Step 3: Plot the Rotated Points

Using the new coordinates \( P'(2, -3) \), \( E'(3, -2) \), \( W'(-1, -1) \), and \( H'(0, 1) \), plot these points on the grid and connect them to form the rotated figure.

Final Answer

The coordinates of the vertices after a \( 90^\circ \) counterclockwise rotation about the origin are:

• \( P(-3, -2)

ightarrow P'(2, -3) \)

• \( E(-2, -3)

ightarrow E'(3, -2) \)

• \( W(-1, 1)

ightarrow W'(-1, -1) \)

• \( H(1, 0)

ightarrow H'(0, 1) \)

(Plot these points to visualize the rotated figure.)