Final answer:
To find the equation of the line through the perpendicular bisector of the line segment connecting (4,2) and (8,4), we first find the midpoint and slope of the line segment. Then, we use the point-slope form of a linear equation to find the equation of the line. The equation of the line is y = 1/2x - 3/2.
Step-by-step explanation:
To find the equation of the line passing through the perpendicular bisector of the line segment connecting (4,2) and (8,4), we first need to find the midpoint of the line segment. The midpoint is the average of the x-coordinates and the average of the y-coordinates. Therefore, the midpoint is ( (4+8)/2, (2+4)/2 ) = (6,3).
Now, we need to find the slope of the line segment connecting (4,2) and (8,4). The slope can be found using the formula: m = (change in y) / (change in x). The change in y is 4-2 = 2 and the change in x is 8-4 = 4. Therefore, the slope of the line segment is m = 2/4 = 1/2.
Since the perpendicular bisector passes through the midpoint, we can use the point-slope form of a linear equation to find the equation of the line. The point-slope form is y - y1 = m(x - x1), where (x1, y1) is the midpoint and m is the slope. Plugging in (6,3) for (x1, y1) and 1/2 for m, we get y - 3 = 1/2(x - 6).
Simplifying the equation gives us the equation of the line through the perpendicular bisector: y = 1/2x - 3/2.