Question

1. for the figure shown here, a. rotate segment cd 180° around point d. b. rotate segment cd 180° around point e. c. rotate segment cd 180° around point m.

Explanation:

Step1: Recall rotation rules

A 180 - degree rotation around a point $(x_0,y_0)$ of a point $(x,y)$ is given by the transformation $(x,y)\to(2x_0 - x,2y_0 - y)$. For a line - segment, we apply this rule to its endpoints.

Step2: a. Rotate segment CD 180° around point D

Let the coordinates of $C=(x_1,y_1)$ and $D=(x_2,y_2)$. When rotating segment $CD$ 180° around $D$, the image of $D$ is itself, i.e., $D' = D$. The image of $C$, $C'$ has coordinates $(2x_2 - x_1,2y_2 - y_1)$. The new segment $C'D$ is a straight - line segment where $D$ is the mid - point of $CC'$.

Step3: b. Rotate segment CD 180° around point E

Let $E=(x_3,y_3)$. For point $C$, its image $C''$ after 180 - degree rotation around $E$ has coordinates $(2x_3 - x_1,2y_3 - y_1)$ and for point $D$, its image $D''$ has coordinates $(2x_3 - x_2,2y_3 - y_2)$. The new segment $C''D''$ is obtained by connecting $C''$ and $D''$.

Step4: c. Rotate segment CD 180° around point M

Let $M=(x_4,y_4)$. For point $C$, its image $C'''$ has coordinates $(2x_4 - x_1,2y_4 - y_1)$ and for point $D$, its image $D'''$ has coordinates $(2x_4 - x_2,2y_4 - y_2)$. The new segment $C'''D'''$ is the result of the rotation.
However, without specific coordinates, we can describe the geometric results:

• a. The new segment has $D$ as one of its endpoints and the other endpoint is on the extension of $CD$ such that $D$ is the mid - point of the original and new endpoints of the segment.
• b. The segment $CD$ is rotated around an external point $E$, creating a new segment with endpoints that are symmetric to $C$ and $D$ with respect to $E$.
• c. Since $M$ is on $CD$, rotating $CD$ 180° around $M$ results in a segment that lies on the same line as $CD$ with $M$ being the mid - point of the segment formed by the endpoints of the original and rotated segments.

Since no specific coordinates are given and we are just asked for a general geometric description of the rotations:

Answer:

a. The new segment has $D$ as one endpoint and the other endpoint is on the line $CD$ such that $D$ is the mid - point between $C$ and its image.
b. The new segment is obtained by finding the points symmetric to $C$ and $D$ with respect to point $E$ and connecting them.
c. The new segment lies on the same line as $CD$ with $M$ being the mid - point of the segment formed by the endpoints of the original and rotated segments.