When rotating a point 90 degrees counterclockwise, which transformation occurs?

When rotating a point 90 degrees counterclockwise, which transformation occurs?

• A. (x,y) = (-x,-y)
• B. (x,y) = (-y,x)
• C. (x,y) = (y,x)
• D. (x,y) = (x,-y)

Answer: (B)

When a point \( (x, y) \) is rotated 90 degrees counterclockwise about the origin, the transformation that occurs can be represented by the equation \( (x, y) = (-y, x) \). This transformation corresponds to the geometric rotation in the coordinate system. To understand this, consider how the point \( (x, y) \) changes position during a 90-degree counterclockwise rotation. The x-coordinate of the original point becomes the y-coordinate in the new position, and the y-coordinate of the original point becomes the negative of the x-coordinate in the new position. This effectively moves the point from its original quadrant to the appropriate new quadrant in the counterclockwise direction, thus arriving at \( (-y, x) \). In this transformation, the original coordinates are altered in a way that maintains the distance from the origin while changing their angular position. This is a fundamental property of rotational transformations in a two-dimensional plane. Other transformations that are presented do not correctly represent a 90-degree counterclockwise rotation. They either reflect the points across axes or swap coordinates without the necessary sign change, which would not yield the positional change required by such a rotation.

When a point ( (x, y) ) is rotated 90 degrees counterclockwise about the origin, the transformation that occurs can be represented by the equation ( (x, y) = (-y, x) ). This transformation corresponds to the geometric rotation in the coordinate system.

To understand this, consider how the point ( (x, y) ) changes position during a 90-degree counterclockwise rotation. The x-coordinate of the original point becomes the y-coordinate in the new position, and the y-coordinate of the original point becomes the negative of the x-coordinate in the new position. This effectively moves the point from its original quadrant to the appropriate new quadrant in the counterclockwise direction, thus arriving at ( (-y, x) ).

In this transformation, the original coordinates are altered in a way that maintains the distance from the origin while changing their angular position. This is a fundamental property of rotational transformations in a two-dimensional plane.

Other transformations that are presented do not correctly represent a 90-degree counterclockwise rotation. They either reflect the points across axes or swap coordinates without the necessary sign change, which would not yield the positional change required by such a rotation.