Question

Point (6,7) is rotated 540° counterclockwise about the origin. What is the x-coordinate of the point after this rotation? A) 6

B) -6
C) 7
D) -7

Answer 1

A 540° counterclockwise rotation of the point (6,7) results in a point directly opposite the original, so the x-coordinate becomes -6.

Step-by-step explanation:

When the point (6,7) is rotated 540° counterclockwise about the origin, it is equivalent to a rotation of 540° - 360° = 180° counterclockwise, since rotating 360° brings the point back to its original position. A 180° rotation will place point (6,7) directly opposite its original position on the cartesian plane, flipping it over both axes. Thus, the new x-coordinate will be the negative of the original x-coordinate, which is -6. Therefore, the correct answer is B) -6.

Answer 2

B) -6

After a 540° counterclockwise rotation about the origin, the x-coordinate of point (6,7) becomes -6, reflecting the geometric effect of the transformation.

Step-by-step explanation:

After rotating the point (6,7) 540° counterclockwise about the origin, the x-coordinate becomes -6. To understand this transformation, we can break down the rotation angle. A 540° counterclockwise rotation is equivalent to one and a half revolutions. During each revolution, the x-coordinate of a point is inverted (multiplied by -1) due to the change in direction.

Initially, the x-coordinate is 6. After one revolution, it becomes -6, and after the additional half revolution, it remains -6. The y-coordinate may also change, but since we're interested in the x-coordinate, it's important to note that it becomes -6 after the described rotation.

This geometric interpretation aligns with the trigonometric understanding of rotations and is crucial for visualizing the effect of transformations on points in the Cartesian plane.