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Which rule represents [tex]R_{0,180^{-}}[/tex]?

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

Sagot :

GinnyAnswer
To determine which rule represents the transformation \( R_{0,180^\circ} \), we need to understand what a 180-degree rotation around the origin entails. A 180-degree rotation around the origin changes a point \((x, y)\) to its diametrically opposite position.

Visualizing this on the Cartesian plane, a point \((x, y)\) would end up at \((-x, -y)\) when rotated 180 degrees around the origin. This is because you effectively move the point to the opposite quadrant each rotation.

Let's verify this with an example:
- Consider a point \((3, 4)\). When rotated 180 degrees, it becomes \((-3, -4)\).
- Similarly, a point \((-2, 5)\) becomes \((2, -5)\) when rotated 180 degrees.

Thus, the rule that represents \( R_{0,180^\circ} \) is:
[tex]\[ (x, y) \rightarrow (-x, -y) \][/tex]

So, the correct option is:

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