Final answer:
The value of b for the perpendicular bisector of the line segment from (0,3) to (6,9), represented by the equation xy = b, is 9.
Step-by-step explanation:
The graph of the line x y = b is a perpendicular bisector of the line segment from (0,3) to (6,9). To find the value of b, let's first determine the midpoint of the segment, which the perpendicular bisector will pass through. The midpoint, M, is found using the average of the x-coordinates and the y-coordinates of the endpoints: M = ((0+6)/2, (3+9)/2) = (3, 6).
Now, the slope of the original line segment is (9-3)/(6-0) = 6/6 = 1. Since the desired line is perpendicular to this segment, its slope will be the negative reciprocal of 1, which is -1. Hence, the equation of the bisector will be of the form y = mx + c, where m = -1. Plugging in the midpoint coordinates (3, 6) to find c, we get 6 = -1(3) + c, leading to c = 9.
Therefore, the equation of the bisector is y = -x + 9, which can be rewritten as x y = b. This means b = 9.