Question

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.

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the equation of the perpendicular bisector of AC, we need to find the midpoint of AC and the slope of the line AC, then use these values to find the negative reciprocal slope.

1. Midpoint of AC:
The midpoint formula is given by:
(midpoint_x, midpoint_y) = ((x1 + x2)/2, (y1 + y2)/2)

From the given points A(-2, -2) and C(18, -6), we can calculate the midpoint:
midpoint_x = (-2 + 18)/2 = 16/2 = 8
midpoint_y = (-2 - 6)/2 = -8/2 = -4

The midpoint of AC is (8, -4).

2. Slope of AC:
The slope formula is given by:
slope = (y2 - y1) / (x2 - x1)

From the given points A(-2, -2) and C(18, -6), we can calculate the slope:
slope_AC = (-6 - (-2)) / (18 - (-2)) = (-6 + 2) / (18 + 2) = -4 / 20 = -1/5

3. Negative reciprocal slope:
To find the slope of the line perpendicular to AC, we take the negative reciprocal of the slope_AC:
slope_perpendicular = -1/slope_AC = -1/(-1/5) = 5

Now we have the midpoint (8, -4) and the slope of the perpendicular bisector (5).

4. Equation of the line in point-slope form:
Using the point-slope form of a linear equation, we have:
y - y1 = m(x - x1)

Substituting the values: (x1, y1) = (8, -4) and m = 5, we get:
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.