Final answer:
To perform transformations on points in the coordinate system, we follow different rules for each transformation type, reflecting in the x-axis, rotating 180° around the origin, reflecting in y = x, and rotating -90° around the origin, resulting in new positions for points A, B, and C.
Step-by-step explanation:
Transformations of Points in the Coordinate System
When performing geometric transformations on points in the coordinate system, we follow specific rules depending on the type of transformation. Below are the transformations for points A(6, 4), B(-2, 1), and C(5, 0).
Reflection in the x-axis
To reflect a point in the x-axis, invert the y-coordinate while keeping the x-coordinate the same. Thus:
A' becomes (6, -4)
B' becomes (-2, -1)
C' becomes (5, 0) since it lies on the x-axis
180° Rotation about the origin
A 180° rotation about the origin inverts both coordinates of a point. Therefore:
A'' becomes (-6, -4)
B'' becomes (2, -1)
C'' becomes (-5, 0)
Reflection in y = x
For reflection in the line y = x, swap the x- and y-coordinates of each point. Consequently:
A''' becomes (4, 6)
B''' becomes (1, -2)
C''' becomes (0, 5)
-90° Rotation about the origin
A -90° rotation about the origin (clockwise) involves swapping the coordinates and changing the sign of the original x-coordinate. This results in:
A'''' becomes (4, -6)
B'''' becomes (1, 2)
C'''' becomes (0, -5)