Question

name practice for exercises 1 and 2, use the figure to answer the questions. 1. where will the figure be after it is rotated 90° clockwise about p? draw the image and label it “a”. 2. where will the figure be after it is rotated 180° about p? draw the image and label it “b”.

Explanation:

Problem 1: Rotation 90° Clockwise about \( P \)

To determine the position after a 90° clockwise rotation about point \( P \):

1. Understand Rotation Rules: A 90° clockwise rotation about a point \((x,y)\) transforms a point \((a,b)\) relative to \( P \) as follows: if the vector from \( P \) to the point is \((h,k)\), the new vector after rotation is \((k, -h)\). For the given figure (with a right angle at \( P \)), each segment and the heart - shaped figure will rotate 90° clockwise around \( P \).
2. Visualize the Rotation: The original figure has a heart connected by a segment to \( P \), and another heart and segment forming a right angle. When rotating 90° clockwise about \( P \), the horizontal segment (assuming the original has a horizontal and vertical segment meeting at \( P \) at a right angle) will become vertical, and the vertical segment will become horizontal, with the hearts moving accordingly. The new position (image \( A \)) will have the hearts and segments arranged such that the angle between the segments is still a right angle, but rotated 90° clockwise around \( P \).

Problem 2: Rotation 180° about \( P \)

To determine the position after a 180° rotation about point \( P \):

1. Understand Rotation Rules: A 180° rotation about a point \((x,y)\) transforms a point \((a,b)\) relative to \( P \) such that the vector from \( P \) to the point \((h,k)\) becomes \((-h, -k)\). For the given figure, each point of the figure (hearts and segments) will be rotated 180° around \( P \).
2. Visualize the Rotation: The original figure has two hearts and two segments meeting at \( P \) at a right angle. After a 180° rotation, the segments will be in the opposite direction (collinear with their original paths but extended in the opposite sense), and the hearts will be on the opposite side of \( P \) from their original positions. The new position (image \( B \)) will be such that the figure is symmetric to the original figure with respect to point \( P \) (central symmetry about \( P \)).

Since this is a drawing - based problem, the following is a description of how to draw the images:

Drawing for Problem 1 (90° Clockwise about \( P \)):

1. Identify the two segments meeting at \( P \) (let's say one is horizontal and one is vertical, with the right angle at \( P \)).
2. Rotate the horizontal segment 90° clockwise around \( P \) (it will become a vertical segment pointing downwards if the original horizontal segment was pointing to the right).
3. Rotate the vertical segment 90° clockwise around \( P \) (it will become a horizontal segment pointing to the right if the original vertical segment was pointing upwards).
4. Move the hearts attached to these segments to their new positions according to the rotated segments. Label this new figure as \( A \).

Drawing for Problem 2 (180° about \( P \)):

1. For each segment and heart in the original figure, find the point on the line through \( P \) and the original point such that \( P \) is the mid - point between the original point and the new point.
2. For example, if a heart is at a distance \( d \) from \( P \) along a certain direction, place the new heart (for image \( B \)) at a distance \( d \) from \( P \) in the opposite direction.
3. Adjust the segments accordingly (they will be straight lines passing through \( P \) and extending to the new positions of the hearts) and label the new figure as \( B \).

Since this is a drawing - based task, the final answer is the correctly drawn figures \( A \) (after 90° clockwise rotation about \( P \)) and \( B \) (after 180° rotation about \( P \)) as described above.

Answer:

To determine the position after a 180° rotation about point \( P \):

1. Understand Rotation Rules: A 180° rotation about a point \((x,y)\) transforms a point \((a,b)\) relative to \( P \) such that the vector from \( P \) to the point \((h,k)\) becomes \((-h, -k)\). For the given figure, each point of the figure (hearts and segments) will be rotated 180° around \( P \).
2. Visualize the Rotation: The original figure has two hearts and two segments meeting at \( P \) at a right angle. After a 180° rotation, the segments will be in the opposite direction (collinear with their original paths but extended in the opposite sense), and the hearts will be on the opposite side of \( P \) from their original positions. The new position (image \( B \)) will be such that the figure is symmetric to the original figure with respect to point \( P \) (central symmetry about \( P \)).

Since this is a drawing - based problem, the following is a description of how to draw the images:

Drawing for Problem 1 (90° Clockwise about \( P \)):

1. Identify the two segments meeting at \( P \) (let's say one is horizontal and one is vertical, with the right angle at \( P \)).
2. Rotate the horizontal segment 90° clockwise around \( P \) (it will become a vertical segment pointing downwards if the original horizontal segment was pointing to the right).
3. Rotate the vertical segment 90° clockwise around \( P \) (it will become a horizontal segment pointing to the right if the original vertical segment was pointing upwards).
4. Move the hearts attached to these segments to their new positions according to the rotated segments. Label this new figure as \( A \).

Drawing for Problem 2 (180° about \( P \)):

1. For each segment and heart in the original figure, find the point on the line through \( P \) and the original point such that \( P \) is the mid - point between the original point and the new point.
2. For example, if a heart is at a distance \( d \) from \( P \) along a certain direction, place the new heart (for image \( B \)) at a distance \( d \) from \( P \) in the opposite direction.
3. Adjust the segments accordingly (they will be straight lines passing through \( P \) and extending to the new positions of the hearts) and label the new figure as \( B \).

Since this is a drawing - based task, the final answer is the correctly drawn figures \( A \) (after 90° clockwise rotation about \( P \)) and \( B \) (after 180° rotation about \( P \)) as described above.