Question

rotations
a rotation is a transformation that is a turn
a figure is rotated about a point called the center of rotation
the image has the same shape and size as the pre - image.
the angle of rotation is the number of degrees the figure rotates.
the figure can be rotated clockwise or counter clockwise.
one way to rotate a figure is to visualize the rotation.
the figure below is rotated 90° counterclockwise about the origin. what are the
coordinates of the preimage and image?
vertices of preimage vertices of image
a(1,1) a(-1,1)
b(1,4) b(-4,1)
c(3,4) c(-4,3)
d(3,1) d(-1,3)
you can also algebraically rotate a figure.
rotation image coordinates
clockwise rotation of 90° (x,y)→(y, - x)
counterclockwise rotation of 90° (x,y)→(-y,x)
rotation of 180° (x,y)→(-x, - y)
clockwise rotation of 270° (x,y)→(y,x)
counterclockwise rotation of 270° (x,y)→(-y, - x)
rotate each figure as described.
180° about the origin 270° counterclockwise about the origin 270° clockwise about the origin
coordinates of the image: coordinates of the image: coordinates of the image:

Explanation:

Step1: Recall 180 - degree rotation rule

For a 180 - degree rotation about the origin, the rule is $(x,y)\to(-x,-y)$.

Step2: Apply rule to each vertex

Let's assume the vertices of the pre - image of the parallelogram are $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$. After 180 - degree rotation, the vertices of the image will be $(-x_1,-y_1),(-x_2,-y_2),(-x_3,-y_3),(-x_4,-y_4)$.

Step3: Recall 270 - degree counter - clockwise rotation rule

The rule for a 270 - degree counter - clockwise rotation about the origin is $(x,y)\to(y, - x)$.

Step4: Apply 270 - degree counter - clockwise rule to vertices of pentagon

If the vertices of the pre - image of the pentagon are $(a_1,b_1),(a_2,b_2),(a_3,b_3),(a_4,b_4),(a_5,b_5)$, the vertices of the image will be $(b_1,-a_1),(b_2,-a_2),(b_3,-a_3),(b_4,-a_4),(b_5,-a_5)$.

Step5: Recall 270 - degree clockwise rotation rule

The rule for a 270 - degree clockwise rotation about the origin is $(x,y)\to(-y,x)$.

Step6: Apply 270 - degree clockwise rule to vertices of triangle

If the vertices of the pre - image of the triangle are $(m_1,n_1),(m_2,n_2),(m_3,n_3)$, the vertices of the image will be $(-n_1,m_1),(-n_2,m_2),(-n_3,m_3)$.

Since we don't have the actual coordinates of the pre - images in the problem statement (just the general tasks of rotation), we can't give the exact numerical answers for the coordinates of the images. But the general methods for finding the coordinates of the images after rotation are as described above using the rotation rules.

Answer:

For 180 - degree rotation about the origin: Use the rule $(x,y)\to(-x,-y)$ for each vertex of the pre - image.
For 270 - degree counter - clockwise rotation about the origin: Use the rule $(x,y)\to(y, - x)$ for each vertex of the pre - image.
For 270 - degree clockwise rotation about the origin: Use the rule $(x,y)\to(-y,x)$ for each vertex of the pre - image.