As a domain expert in geometry, I can provide you with a comprehensive understanding of the rules governing rotations in a two-dimensional plane, specifically focusing on a 270-degree rotation about the origin.

Rotations in a two-dimensional plane are transformations that turn a figure around a fixed point without changing its size or shape. The fixed point is known as the center of rotation, and in this case, it is the origin of the coordinate system, which is the point (0,0).

### Understanding Rotations
When we talk about a rotation, we need to consider the following aspects:

1. Center of Rotation: The point around which the rotation occurs.

2. Angle of Rotation: The measure in degrees of how far the figure is turned.

3. Direction of Rotation: Whether the rotation is clockwise or counterclockwise.

### The Rule for a 270-Degree Rotation
The general rule for rotating a point \( (x, y) \) about the origin by an angle \( \theta \) is to use the transformation:
\[ (x, y) \rightarrow (\pm x \cos \theta \pm y \sin \theta) \]
where the signs depend on the quadrant in which the point lies and the direction of rotation (clockwise or counterclockwise).

For a 270-degree rotation, the transformation simplifies because the cosine and sine of 270 degrees are -1 and 0, respectively. However, the sign of the sine function depends on the direction of rotation. If we are considering a counterclockwise rotation, the rule becomes:
\[ (x, y) \rightarrow (y, -x) \]

### Applying the Rule to a Specific Point
Let's apply this rule to the point \( (3, -5) \). According to the rule for a 270-degree counterclockwise rotation about the origin, we get:
\[ (3, -5) \rightarrow (-5, 3) \]

This is because, for a point in the fourth quadrant (where \( x > 0 \) and \( y < 0 \)), the \( y \) coordinate of the new point is the original \( x \), and the \( x \) coordinate is the negation of the original \( y \).

### Clarification on the 180-Degree Rotation Rule
The mention of the 180-degree rotation rule in your question seems to be a point of confusion. The rule for a 180-degree rotation is different and is given by:
\[ (x, y) \rightarrow (-x, -y) \]
Applying this to the point \( (2, 1) \) would indeed yield \( (-2, -1) \), but this is not directly related to the 270-degree rotation we are discussing.

### Summary
To summarize, the rule for a 270-degree rotation about the origin is to take the point \( (x, y) \) and transform it to \( (y, -x) \) for a counterclockwise rotation. This transformation flips the point over both the \( x \)-axis and the \( y \)-axis, effectively rotating it three-quarters of the way around the origin.

Now, let's proceed with the next steps as per your instructions.

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'Using notation, rotate the point (3, -5) about the origin 270 degrees.' Since the rule for a 270 degrees rotation is (x, y) --------> (y, -x), the new point would be (-5, -3). By using the rule for a 180 degrees rotation, we can get the coordinates for the image: (2, 1) becomes (-2, -1)read more >>