Final answer:
The distance from the origin of a point is invariant under rotations because rotations are isometries which preserve distances. Even after rotating point (-6,1), the calculated distance using the sum of the squares of its coordinates remains unchanged. Translation does not affect the distance from the origin.
Step-by-step explanation:
The transformation described, r (rotation) y=x, and T^(2),5 (translation), involves two steps. First, the point is rotated around the origin and then it is translated (shifted). To demonstrate that rotations do not change the distance of a point from the origin, consider point P with coordinates (-6,1). When rotated, the distance from the origin in a coordinate system remains unchanged because a rotation is a type of isometry, which preserves distances. This can be shown mathematically:
The original distance d of point P from the origin is sqrt((-6)^² + (1)^²).
After a rotation, the coordinates of P might change, but the distance will still be calculated by squaring the new coordinates and taking the square root: sqrt((x^²) + (y^²)). Because the sum of the squares of the coordinates remains the same, d does not change.
The transformation T^(2),5 would then translate the point by adding 2 to the x-coordinate and 5 to the y-coordinate. However, since the question asks specifically about the rotation's effect on distance from the origin, it is the rotation that is key in showing that d is invariant.