Given ΔGHI with G(4, -3), H(-4, 2), and I(2, 4), find the perpendicular bisector of HI in standard form.

Answer 1

To find the equation of the perpendicular bisector of HI, we need to find its slope and a point that lies on the line.

Step 1: Find the slope of HI.
The slope of HI is equal to the change in y divided by the change in x:

slope of HI = (yI - yH)/(xI - xH) = (4 - 2)/(2 - (-4)) = 2 / 6 = 1 / 3

Step 2: Find the midpoint of HI.
The midpoint of a line segment is the average of the x-coordinates and the average of the y-coordinates of the endpoints. The coordinates of the midpoint of HI are:

x-coordinate of midpoint = (xH + xI)/2 = (-4 + 2)/2 = -2/2 = -1
y-coordinate of midpoint = (yH + yI)/2 = (2 + 4)/2 = 6/2 = 3

So the midpoint of HI is (-1, 3).

Step 3: Find the slope of the perpendicular bisector.
The slope of the perpendicular bisector is the negative reciprocal of the slope of HI:

slope of perpendicular bisector = -1 / (1/3) = -3

Step 4: Use the midpoint and the slope of the perpendicular bisector to find the equation of the line.
Using the point-slope form of a linear equation, y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope, we can substitute (-1, 3) for (x1, y1) and -3 for m:

y - 3 = -3(x - (-1))

Simplifying, we have:

y - 3 = -3(x + 1)
y - 3 = -3x - 3
y = -3x

So the equation of the perpendicular bisector of HI in standard form is y = -3x.