Triangle ABC has the following points: A (-2,-2), B (4,4), C (18,-6). Use these points to write the equations of the line containing the perpendicular bisector of AC in point slope form. Make sure to show all work in order to receive full points.
To find the midpoint of line segment AC, we average the x-coordinates and the y-coordinates of points A and C separately.
Midpoint's x-coordinate = (x-coordinate of A + x-coordinate of C) / 2
Midpoint's x-coordinate = (-2 + 18) / 2
Midpoint's x-coordinate = 16 / 2
Midpoint's x-coordinate = 8
Midpoint's y-coordinate = (y-coordinate of A + y-coordinate of C) / 2
Midpoint's y-coordinate = (-2 + -6) / 2
Midpoint's y-coordinate = -8 / 2
Midpoint's y-coordinate = -4
Therefore, the midpoint of line segment AC is (8, -4).
The slope of AC can be found using the slope formula:
Slope of AC = (y-coordinate of C - y-coordinate of A) / (x-coordinate of C - x-coordinate of A)
Slope of AC = (-6 - -2) / (18 - -2)
Slope of AC = (-6 + 2) / (18 + 2)
Slope of AC = -4 / 20
Slope of AC = -1/5
Since the perpendicular bisector of a line segment is the line that is perpendicular to it and passes through its midpoint, we can find the slope of the perpendicular bisector by taking the negative reciprocal of the slope of AC.
Slope of perpendicular bisector = -1 / (slope of AC)
Slope of perpendicular bisector = -1 / (-1/5)
Slope of perpendicular bisector = 5
Using the slope-intercept form of a line (y = mx + b), we can write the equation of the line containing the perpendicular bisector of AC.
y - y1 = m(x - x1)
y - (-4) = 5(x - 8)
y + 4 = 5x - 40
y = 5x - 40 - 4
y = 5x - 44
Therefore, the equation of the line containing the perpendicular bisector of AC in point-slope form is y = 5x - 44.