A pentagon is transformed according to the rule [tex][tex]$R_{0,180^{\circ}}$[/tex][/tex]. Which is another way to state the transformation?

A. [tex][tex]$(x, y) \rightarrow(-x, -y)$[/tex][/tex]
B. [tex][tex]$(x, y) \rightarrow(-y, -x)$[/tex][/tex]
C. [tex][tex]$(x, y) \rightarrow(x, -y)$[/tex][/tex]
D. [tex][tex]$(x, y) \rightarrow(-x, y)$[/tex][/tex]



Answer :

To determine the correct transformation of a pentagon according to the rule [tex]\( R_{0,180^{\circ}} \)[/tex], we need to understand what this rule represents.

The rule [tex]\( R_{0,180^{\circ}} \)[/tex] means we are rotating the point around the origin by 180 degrees. When you rotate a point [tex]\((x, y)\)[/tex] by 180 degrees around the origin, both the x-coordinate and the y-coordinate change signs. Specifically, the point [tex]\((x, y)\)[/tex] will move to [tex]\((-x, -y)\)[/tex].

Given the options:

1. [tex]\((x, y) \rightarrow (-x, -y)\)[/tex]
2. [tex]\((x, y) \rightarrow (-y, -x)\)[/tex]
3. [tex]\((x, y) \rightarrow (x, -y)\)[/tex]
4. [tex]\((x, y) \rightarrow (-x, y)\)[/tex]

By definition of the 180-degree rotation about the origin:

- Option 1: [tex]\((x, y) \rightarrow (-x, -y)\)[/tex] depicts the transformation accurately. Both coordinates change their signs, which is what happens during a 180-degree rotation around the origin.
- Option 2: [tex]\((x, y) \rightarrow (-y, -x)\)[/tex] does not correctly represent a 180-degree rotation, as the coordinates are swapped and their signs are changed.
- Option 3: [tex]\((x, y) \rightarrow (x, -y)\)[/tex] only changes the sign of the y-coordinate, which represents a reflection over the x-axis rather than a rotation.
- Option 4: [tex]\((x, y) \rightarrow (-x, y)\)[/tex] only changes the sign of the x-coordinate, which represents a reflection over the y-axis rather than a rotation.

Thus, the correct transformation according to [tex]\( R_{0,180^{\circ}} \)[/tex] is:

[tex]\[ (x, y) \rightarrow (-x, -y) \][/tex]

Therefore, the answer is:

[tex]\((x, y) \rightarrow (-x, -y)\)[/tex]