Final answer:
The equation of the perpendicular bisector to the line passing through the points (2,8) and (4,0) is not listed in the provided options. After calculating the slope of the original line, finding the midpoint. and applying the point-slope form, the equation is determined to be y = 1/4x + 9/4, which is not an option given.
Step-by-step explanation:
The question asks for the equation of the perpendicular bisector to the line passing through the given points (2,8) and (4,0). First, we need to determine the slope of the original line using the formula (y2 - y1) / (x2 - x1). Then, we find the midpoint of the line segment joining the points, which would be a point on the perpendicular bisector. Next, we find the negative reciprocal of the original slope since the slope of the perpendicular bisector is the negative reciprocal of the original line. Finally, we use the point-slope form of a line to construct the equation of the perpendicular bisector.
The original slope is calculated as (0 - 8) / (4 - 2) = -8 / 2 = -4. So, the slope of the perpendicular bisector will be 1/4, since it is the negative reciprocal of -4. The midpoint of the original line segment is ((2+4)/2, (8+0)/2) = (3, 4). Using the point-slope form of the line y - y1 = m(x - x1), where m is the slope, and (x1, y1) is the midpoint, we have y - 4 = 1/4(x - 3). Rearrange this to solve for y, we get y = 1/4x + 3 - 3/4 which simplifies to y = 1/4x + 2.25 or y = 1/4x + 9/4. Therefore, none of the options a, b, c, or d match.