Final answer:
The slope for the line passing through the points (1, 0.1) and (7, 26.8) is 4.5. The distance of point P to the origin remains unchanged under coordinate system rotations because the standard distance formula is invariant with respect to rotations, relying only on the sums of the squares of the coordinates.
Step-by-step explanation:
To find the slope of a line passing through two points, you can use the formula for slope, which is (y2 - y1)/(x2 - x1). Applying this to the given points (1, 0.1) and (7, 26.8), we get:
slope = (26.8 - 0.1) / (7 - 1)
slope = 26.7 / 6
slope = 4.45
Therefore, the slope of the line is 4.5, which corresponds to option b. Note that the question asks for a close approximation, hence why the calculated slope of 4.45 is matched with option b, which is 4.5.
To confirm that the distance of point P to the origin is invariant under rotations of the coordinate system, you need to use the invariance property of the distance formula in a Cartesian coordinate system. The standard distance formula in coordinates is:
√(x2 + y2).
Since the formula only involves the squares of the coordinates, it remains unchanged even if the coordinates of point P are rotated, as the sums of the squares of the coordinates will be the same.