Question

now, lets investigate rotating figures more than 180° about the origin.

1. consider the parallelogram shown on the

coordinate plane.
a. place patty paper on the coordinate
plane, trace the parallelogram, and then
copy the labels for the vertices.
b. rotate the figure 270° clockwise about
the origin. then, identify the coordinates
of the rotated figure and draw the rotated
figure on the coordinate plane.
c. complete the table with the coordinates
of the pre-image and the image.
table with coordinates of pre-image and coordinates of image
d. compare the coordinates of the image with the coordinates of
the pre-image. how are the values of the coordinates the same?
how are they different? explain your reasoning.

Explanation:

Part a

This is a practical step. Place the patty paper over the coordinate plane, carefully trace the parallelogram (including its vertices) and transfer the labels of the vertices (let's assume the parallelogram has vertices, for example, let's identify the vertices from the grid: looking at the right parallelogram, let's say \( N(3,1) \), \( P(3,3) \), \( Q(6,3) \), \( R(6,1) \) – we'll confirm with rotation rules later).

Part b

Step 1: Recall the rotation rule for \( 270^\circ \) clockwise about the origin

The rule for rotating a point \( (x,y) \) \( 270^\circ \) clockwise about the origin is \( (x,y) \to (y, -x) \).

Step 2: Identify pre - image vertices

From the coordinate plane (right parallelogram), let's assume the vertices of the pre - image (parallelogram \( NPCR \) or similar) are:

• Let \( N=(3,1) \)
• Let \( P=(3,3) \)
• Let \( C=(6,3) \)
• Let \( R=(6,1) \)

Step 3: Apply the rotation rule to each vertex

• For \( N=(3,1) \): Using \( (x,y)\to(y, - x) \), we get \( (1,-3) \)
• For \( P=(3,3) \): Using \( (x,y)\to(y, - x) \), we get \( (3,-3) \)
• For \( C=(6,3) \): Using \( (x,y)\to(y, - x) \), we get \( (3,-6) \)
• For \( R=(6,1) \): Using \( (x,y)\to(y, - x) \), we get \( (1,-6) \)

Then, plot these new points \( (1,-3) \), \( (3,-3) \), \( (3,-6) \), \( (1,-6) \) on the coordinate plane and connect them to form the rotated parallelogram.

Part c

Coordinates of Pre - Image	Coordinates of Image
\( (3,3) \)	\( (3,-3) \)
\( (6,3) \)	\( (3,-6) \)
\( (6,1) \)	\( (1,-6) \)

Part d

Similarities and Differences in Coordinates

• Same: The absolute values of the coordinates are related. For example, if we consider the \( x \) - coordinate of the pre - image and the \( y \) - coordinate of the image, and the \( y \) - coordinate of the pre - image and the negative of the \( x \) - coordinate of the image, we can see that the numerical values (ignoring sign and axis) have a correspondence. Also, for some points (like \( N \) and \( P \) where \( x \) - coordinate is the same in pre - image), in the image, the \( y \) - coordinate has a relationship.
• Different: The sign of the \( x \) - coordinate of the image is the negative of the \( y \) - coordinate of the pre - image, and the \( y \) - coordinate of the image is the \( x \) - coordinate of the pre - image. Geometrically, the figure has been rotated \( 270^\circ \) clockwise, which changes the orientation of the figure with respect to the axes. The \( x \) and \( y \) coordinates are swapped and the new \( x \) - coordinate (original \( y \)) keeps its sign while the new \( y \) - coordinate (negative of original \( x \)) has its sign flipped. This is due to the rotation rule \( (x,y)\to(y, - x) \) for a \( 270^\circ \) clockwise rotation about the origin, which is derived from the properties of rotational transformations in the coordinate plane (using the unit circle and angle - based coordinate transformations).

Answer:

Part a

This is a practical step. Place the patty paper over the coordinate plane, carefully trace the parallelogram (including its vertices) and transfer the labels of the vertices (let's assume the parallelogram has vertices, for example, let's identify the vertices from the grid: looking at the right parallelogram, let's say \( N(3,1) \), \( P(3,3) \), \( Q(6,3) \), \( R(6,1) \) – we'll confirm with rotation rules later).

Part b

Step 1: Recall the rotation rule for \( 270^\circ \) clockwise about the origin

The rule for rotating a point \( (x,y) \) \( 270^\circ \) clockwise about the origin is \( (x,y) \to (y, -x) \).

Step 2: Identify pre - image vertices

From the coordinate plane (right parallelogram), let's assume the vertices of the pre - image (parallelogram \( NPCR \) or similar) are:

• Let \( N=(3,1) \)
• Let \( P=(3,3) \)
• Let \( C=(6,3) \)
• Let \( R=(6,1) \)

Step 3: Apply the rotation rule to each vertex

• For \( N=(3,1) \): Using \( (x,y)\to(y, - x) \), we get \( (1,-3) \)
• For \( P=(3,3) \): Using \( (x,y)\to(y, - x) \), we get \( (3,-3) \)
• For \( C=(6,3) \): Using \( (x,y)\to(y, - x) \), we get \( (3,-6) \)
• For \( R=(6,1) \): Using \( (x,y)\to(y, - x) \), we get \( (1,-6) \)

Then, plot these new points \( (1,-3) \), \( (3,-3) \), \( (3,-6) \), \( (1,-6) \) on the coordinate plane and connect them to form the rotated parallelogram.

Part c

Coordinates of Pre - Image	Coordinates of Image
\( (3,3) \)	\( (3,-3) \)
\( (6,3) \)	\( (3,-6) \)
\( (6,1) \)	\( (1,-6) \)

Part d

Similarities and Differences in Coordinates

• Same: The absolute values of the coordinates are related. For example, if we consider the \( x \) - coordinate of the pre - image and the \( y \) - coordinate of the image, and the \( y \) - coordinate of the pre - image and the negative of the \( x \) - coordinate of the image, we can see that the numerical values (ignoring sign and axis) have a correspondence. Also, for some points (like \( N \) and \( P \) where \( x \) - coordinate is the same in pre - image), in the image, the \( y \) - coordinate has a relationship.
• Different: The sign of the \( x \) - coordinate of the image is the negative of the \( y \) - coordinate of the pre - image, and the \( y \) - coordinate of the image is the \( x \) - coordinate of the pre - image. Geometrically, the figure has been rotated \( 270^\circ \) clockwise, which changes the orientation of the figure with respect to the axes. The \( x \) and \( y \) coordinates are swapped and the new \( x \) - coordinate (original \( y \)) keeps its sign while the new \( y \) - coordinate (negative of original \( x \)) has its sign flipped. This is due to the rotation rule \( (x,y)\to(y, - x) \) for a \( 270^\circ \) clockwise rotation about the origin, which is derived from the properties of rotational transformations in the coordinate plane (using the unit circle and angle - based coordinate transformations).