How do you rotate counterclockwise?

Hello there, I'm an expert in geometry and transformations. Let's dive into the concept of rotating a point counterclockwise around the origin.

When we talk about rotating a point around the origin, we're essentially performing a transformation on the Cartesian plane. The rotation we're discussing is a counterclockwise rotation, which means that if you were to rotate a point around the origin, you would turn it in the direction opposite to the way the hands of a clock move.

To understand how to rotate a point counterclockwise, let's first define what a rotation is. A rotation is a transformation that turns a figure around a fixed point without changing its size or shape. The fixed point is known as the center of rotation, and in our case, it's the origin (0,0). The amount of rotation is measured in degrees and is called the angle of rotation.

Now, let's discuss the specific rotations you've mentioned:


1. 90-degree counterclockwise rotation: When you rotate a point 90 degrees counterclockwise around the origin, the coordinates of the point are transformed in a specific way. If you have a point A with coordinates (x, y), after a 90-degree counterclockwise rotation, the new coordinates B will be (-y, x). This is because the x-coordinate of the point becomes the y-coordinate of the new point, and the y-coordinate of the original point becomes the negative of the x-coordinate of the new point.

For example, if we have point A with coordinates (5, 2), after a 90-degree counterclockwise rotation, the image of A, which we'll call point B, will have coordinates (-2, 5). This is because the x-coordinate of A (which is 5) becomes the y-coordinate of B, and the y-coordinate of A (which is 2) becomes the negative of the x-coordinate of B, resulting in -2.


2. 180-degree counterclockwise rotation: A 180-degree rotation around the origin is a bit more straightforward. For any point A with coordinates (x, y), after a 180-degree counterclockwise rotation, the new coordinates C will be (-x, -y). This means both the x and y coordinates of the original point are negated.

Using our point A (5, 2) again, after a 180-degree counterclockwise rotation, the image of A, which we'll call point C, will have coordinates (-5, -2). Both the x and y values are simply flipped in sign.

Now, let's apply these rules to the points you've provided:

- Graph A (5, 2): This is our starting point.
- Graph B: The image of A under a 90-degree counterclockwise rotation about the origin. Using the rule we've discussed, the coordinates of B will be (-2, 5).
- Graph C: The image of A under a 180-degree counterclockwise rotation about the origin. Following the rule for a 180-degree rotation, the coordinates of C will be (-5, -2).

To visualize these rotations, you can imagine a square with point A as one of its corners. If you rotate the square 90 degrees counterclockwise, the corner that was A will now be at the position of B. Similarly, a 180-degree rotation would place the corner that was A at the position of C, directly opposite A on the square.

In summary, rotating points counterclockwise around the origin is a matter of understanding how the coordinates change with respect to the angle of rotation. For a 90-degree rotation, you swap the x and y coordinates and negate the new x-coordinate. For a 180-degree rotation, you negate both the x and y coordinates of the original point.

Now, let's move on to the next step.

Graph A(5, 2), then graph B, the image of A under a 90-- counterclockwise rotation about the origin. Rule for 90-- counterclockwise rotation: A (5, 2) B (- 2, 5) Now graph C, the image of A under a 180-- counterclockwise rotation about the origin.