Question

c. rotate 90° clockwise around (-2,1).

Explanation:

To solve the problem of rotating a figure \( 90^\circ \) clockwise around the point \( (-2, 1) \), we follow these steps:

Step 1: Identify the vertices of the original figure

First, we need to determine the coordinates of the vertices of the given triangle (the shaded figure). Let's assume the vertices are \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) (we'll find their coordinates from the grid). From the grid, let's assume the vertices are:

• \( A(-5, 3) \)
• \( B(-3, 5) \)
• \( C(-1, 3) \)
• \( D(-3, 1) \) (Wait, looking at the grid, maybe it's a quadrilateral? Let's correct. Let's re-examine the grid. Let's list the coordinates properly. Let's assume the center of rotation is \( O(-2, 1) \). Let's take the vertices of the shaded figure:

Let’s find the coordinates of each vertex relative to the center of rotation \( O(-2, 1) \).

Step 2: Translate the figure so that the center of rotation is at the origin

To rotate a figure around a point \( (h, k) \), we first translate the figure so that \( (h, k) \) becomes \( (0, 0) \). The translation rule is: \( (x, y) \to (x - h, y - k) \). Here, \( h = -2 \), \( k = 1 \), so the translation is \( (x, y) \to (x + 2, y - 1) \) (since \( x - (-2) = x + 2 \), \( y - 1 = y - 1 \)).

Let’s apply this to each vertex:

1. Let’s assume the first vertex \( P_1(-5, 3) \):

• Translated: \( (-5 + 2, 3 - 1) = (-3, 2) \)

1. Second vertex \( P_2(-3, 5) \):

• Translated: \( (-3 + 2, 5 - 1) = (-1, 4) \)

1. Third vertex \( P_3(-1, 3) \):

• Translated: \( (-1 + 2, 3 - 1) = (1, 2) \)

1. Fourth vertex \( P_4(-3, 1) \):

• Translated: \( (-3 + 2, 1 - 1) = (-1, 0) \)

Step 3: Rotate the translated vertices \( 90^\circ \) clockwise

The rule for rotating a point \( (x, y) \) \( 90^\circ \) clockwise around the origin is \( (x, y) \to (y, -x) \).

Let’s apply this to each translated vertex:

1. For \( (-3, 2) \):

• Rotated: \( (2, 3) \) (since \( x = -3 \), \( y = 2 \); new \( x = y = 2 \), new \( y = -x = 3 \))

1. For \( (-1, 4) \):

• Rotated: \( (4, 1) \) (new \( x = 4 \), new \( y = -(-1) = 1 \))

1. For \( (1, 2) \):

• Rotated: \( (2, -1) \) (new \( x = 2 \), new \( y = -1 \))

1. For \( (-1, 0) \):

• Rotated: \( (0, 1) \) (new \( x = 0 \), new \( y = -(-1) = 1 \))

Step 4: Translate the rotated vertices back to the original center of rotation

Now, we translate the rotated vertices back by reversing the initial translation. The reverse translation is \( (x, y) \to (x - 2, y + 1) \) (since we initially added 2 to \( x \) and subtracted 1 from \( y \); now we subtract 2 from \( x \) and add 1 to \( y \)).

Let’s apply this to each rotated vertex:

1. For \( (2, 3) \):

• Translated back: \( (2 - 2, 3 + 1) = (0, 4) \)

1. For \( (4, 1) \):

• Translated back: \( (4 - 2, 1 + 1) = (2, 2) \)

1. For \( (2, -1) \):

• Translated back: \( (2 - 2, -1 + 1) = (0, 0) \)

1. For \( (0, 1) \):

• Translated back: \( (0 - 2, 1 + 1) = (-2, 2) \)

Step 5: Plot the new vertices

Now, we plot the new vertices \( (0, 4) \), \( (2, 2) \), \( (0, 0) \), and \( (-2, 2) \) and connect them to get the rotated figure.

Final Answer

The rotated figure has vertices at \( (0, 4) \), \( (2, 2) \), \( (0, 0) \), and \( (-2, 2) \) (or adjust based on the actual original vertices from the grid). The key steps involve translation, rotation, and reverse translation.

Answer:

To solve the problem of rotating a figure \( 90^\circ \) clockwise around the point \( (-2, 1) \), we follow these steps:

Step 1: Identify the vertices of the original figure

First, we need to determine the coordinates of the vertices of the given triangle (the shaded figure). Let's assume the vertices are \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) (we'll find their coordinates from the grid). From the grid, let's assume the vertices are:

• \( A(-5, 3) \)
• \( B(-3, 5) \)
• \( C(-1, 3) \)
• \( D(-3, 1) \) (Wait, looking at the grid, maybe it's a quadrilateral? Let's correct. Let's re-examine the grid. Let's list the coordinates properly. Let's assume the center of rotation is \( O(-2, 1) \). Let's take the vertices of the shaded figure:

Let’s find the coordinates of each vertex relative to the center of rotation \( O(-2, 1) \).

Step 2: Translate the figure so that the center of rotation is at the origin

To rotate a figure around a point \( (h, k) \), we first translate the figure so that \( (h, k) \) becomes \( (0, 0) \). The translation rule is: \( (x, y) \to (x - h, y - k) \). Here, \( h = -2 \), \( k = 1 \), so the translation is \( (x, y) \to (x + 2, y - 1) \) (since \( x - (-2) = x + 2 \), \( y - 1 = y - 1 \)).

Let’s apply this to each vertex:

1. Let’s assume the first vertex \( P_1(-5, 3) \):

• Translated: \( (-5 + 2, 3 - 1) = (-3, 2) \)

1. Second vertex \( P_2(-3, 5) \):

• Translated: \( (-3 + 2, 5 - 1) = (-1, 4) \)

1. Third vertex \( P_3(-1, 3) \):

• Translated: \( (-1 + 2, 3 - 1) = (1, 2) \)

1. Fourth vertex \( P_4(-3, 1) \):

• Translated: \( (-3 + 2, 1 - 1) = (-1, 0) \)

Step 3: Rotate the translated vertices \( 90^\circ \) clockwise

The rule for rotating a point \( (x, y) \) \( 90^\circ \) clockwise around the origin is \( (x, y) \to (y, -x) \).

Let’s apply this to each translated vertex:

1. For \( (-3, 2) \):

• Rotated: \( (2, 3) \) (since \( x = -3 \), \( y = 2 \); new \( x = y = 2 \), new \( y = -x = 3 \))

1. For \( (-1, 4) \):

• Rotated: \( (4, 1) \) (new \( x = 4 \), new \( y = -(-1) = 1 \))

1. For \( (1, 2) \):

• Rotated: \( (2, -1) \) (new \( x = 2 \), new \( y = -1 \))

1. For \( (-1, 0) \):

• Rotated: \( (0, 1) \) (new \( x = 0 \), new \( y = -(-1) = 1 \))

Step 4: Translate the rotated vertices back to the original center of rotation

Now, we translate the rotated vertices back by reversing the initial translation. The reverse translation is \( (x, y) \to (x - 2, y + 1) \) (since we initially added 2 to \( x \) and subtracted 1 from \( y \); now we subtract 2 from \( x \) and add 1 to \( y \)).

Let’s apply this to each rotated vertex:

1. For \( (2, 3) \):

• Translated back: \( (2 - 2, 3 + 1) = (0, 4) \)

1. For \( (4, 1) \):

• Translated back: \( (4 - 2, 1 + 1) = (2, 2) \)

1. For \( (2, -1) \):

• Translated back: \( (2 - 2, -1 + 1) = (0, 0) \)

1. For \( (0, 1) \):

• Translated back: \( (0 - 2, 1 + 1) = (-2, 2) \)

Step 5: Plot the new vertices

Now, we plot the new vertices \( (0, 4) \), \( (2, 2) \), \( (0, 0) \), and \( (-2, 2) \) and connect them to get the rotated figure.

Final Answer

The rotated figure has vertices at \( (0, 4) \), \( (2, 2) \), \( (0, 0) \), and \( (-2, 2) \) (or adjust based on the actual original vertices from the grid). The key steps involve translation, rotation, and reverse translation.