Final answer:
To find the equation of the line through the perpendicular bisector of the line segment connecting (4,2) and (8,4), you first need to find the midpoint of the line segment and then determine the slope of the perpendicular line. Using the point-slope form, you can plug in the coordinates of the midpoint and the negative reciprocal of the slope to find the equation of the line through the perpendicular bisector.
Step-by-step explanation:
To find the equation of the line 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 formula is given by:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Plugging in the coordinates, we get:
Midpoint = ((4 + 8) / 2, (2 + 4) / 2) = (6, 3)
The slope of the line segment connecting (4,2) and (8,4) is:
Slope = (y2 - y1) / (x2 - x1) = (4 - 2) / (8 - 4) = 2/4 = 1/2
The slope of a line perpendicular to this line segment is the negative reciprocal of 1/2, which is -2. Using the point-slope form of a linear equation, we can determine the equation of the line:
y - y1 = m(x - x1)
Substituting the coordinates (6, 3) and the slope -2, we get:
y - 3 = -2(x - 6)
Simplifying, we have:
y - 3 = -2x + 12
Finally, rearranging the equation, we get the correct answer:
y = -2x + 15