Final answer:
The question regarding the x-coordinate of R' in a dilation cannot be answered without additional information. However, the discussion of invariance of distances under rotations of the coordinate system explains that such distances remain constant regardless of the rotation of the coordinate axis in both two-dimensional and three-dimensional space.
Step-by-step explanation:
The student's question seems to contain a typo and lacks clarity on essential details such as the scale factor of dilation and the original coordinates of point R in pentagon MNPQR. However, to provide instructional support on the related topic, we can discuss the invariance of distance in transformations.
When we talk about distance invariance under rotational transformations of the coordinate system, we refer to the fact that distances between points and the origin, or between two points, do not change regardless of how the coordinate system is rotated. This is a fundamental property of rotational transformations in both two-dimensional and three-dimensional space.
For a point P with coordinates (x, y) in two-dimensional space, the distance r from P to the origin (0,0) is r = √(x² + y²), according to the Pythagorean theorem. This distance remains the same after rotation because the new coordinates (x', y') will satisfy the same relationship: r' = √(x'² + y'²) = r.
In three-dimensional space, a point P has coordinates (x, y, z), and its distance to origin is r = √(x² + y² + z²). Under rotation, the distance to the origin remains unchanged even though the position of point P changes.