Final answer:
To find the equation of a line that is perpendicular to y = 4x + 2 and passes through the midpoint of (-3,8) and (5,12), use the midpoint formula to find the coordinates of the midpoint and then use the slope-intercept form of a line to find the equation.
Step-by-step explanation:
To find the equation of a line that is perpendicular to y = 4x + 2 and passes through the midpoint of (-3,8) and (5,12), we need to determine the slope of the line perpendicular to y = 4x + 2. The slope of a line perpendicular to y = mx + b is the negative reciprocal of the original slope. In this case, the slope of y = 4x + 2 is 4, so the slope of the perpendicular line is -1/4.
To find the equation of the perpendicular line, we can use the midpoint formula to find the coordinates of the midpoint. The midpoint formula is (x1 + x2)/2, (y1 + y2)/2. Substituting the values of (-3,8) and (5,12) into the formula, we get the coordinates of the midpoint as (1,10).
Now we can use the slope-intercept form of a line, y = mx + b, where m is the slope and b is the y-intercept. We know that the slope is -1/4, and the line passes through the point (1,10). Substituting these values into the equation, we get y = -1/4x + 12.5 as the equation of the line perpendicular to y = 4x + 2 and passing through the midpoint.