Final answer:
To find the equation of the line perpendicular to the line passing through (3,6) and (-5,0) and passing through the midpoint of those two points, we need to first find the slope of the given line. The slope of a line passing through two points is given by the formula: m = (y2 - y1) / (x2 - x1). The slope of the line perpendicular to this line will be the negative reciprocal of this slope, which is -4/3. Finally, we can use the slope-intercept form of the equation of a line, y = mx + b, where m is the slope and b is the y-intercept.
Step-by-step explanation:
To find the equation of the line perpendicular to the line passing through (3,6) and (-5,0) and passing through the midpoint of those two points, we need to first find the slope of the given line. The slope of a line passing through two points is given by the formula: m = (y2 - y1) / (x2 - x1). Substituting the values, we get m = (0 - 6) / (-5 - 3) = -6 / -8 = 3/4. The slope of the line perpendicular to this line will be the negative reciprocal of this slope, which is -4/3.
Now, we need to find the midpoint of the points (3,6) and (-5,0). The coordinates of the midpoint can be found using the formula: (x, y) = ((x1 + x2) / 2, (y1 + y2) / 2). Substituting the values, we get: (x, y) = ((3 + -5) / 2, (6 + 0) / 2) = (-1, 3).
Finally, we can use the slope-intercept form of the equation of a line, y = mx + b, where m is the slope and b is the y-intercept. We know the slope is -4/3 and the line passes through the point (-1, 3). Substituting these values, we get y = (-4/3)x + b. To find b, we can substitute the coordinates of the points (-1, 3) into the equation and solve for b. Using the equation 3 = (-4/3)(-1) + b, we can simplify and find that b = 1.
Therefore, the equation of the line perpendicular to the line passing through (3,6) and (-5,0) and passing through the midpoint of those two points is y = (-4/3)x + 1.