given triangle GHI with G(4,-3), H(-4,2) and I(2,4), find the perpendicular bisector of HI in standard form.
To find the perpendicular bisector of HI, we need to find the midpoint of HI and then determine the slope of the perpendicular bisector.
First, let's find the midpoint of HI. The formula to find the midpoint of two points (x1, y1) and (x2, y2) is:
Midpoint = [(x1 + x2)/2, (y1 + y2)/2]
So, using the coordinates of H(-4, 2) and I(2, 4):
Midpoint = [((-4 + 2)/2), ((2 + 4)/2)]
Midpoint = [-1, 3]
Now, let's find the slope of HI using the formula for slope:
Slope of HI = (y2 - y1)/(x2 - x1)
Using the coordinates of H(-4, 2) and I(2, 4):
Slope of HI = (4 - 2)/(2 - (-4))
Slope of HI = 2/6 = 1/3
Since the perpendicular bisector of a line has a slope that is the negative reciprocal of the original line, the slope of the perpendicular bisector will be -3.
Now, we can use the point-slope form of a line (y - y1) = m(x - x1), with m = -3 and (x1, y1) = (-1, 3) to find the equation of the perpendicular bisector.
Using the point (-1, 3) and the slope -3, the equation of the perpendicular bisector of HI is:
y - 3 = -3(x - (-1))
Simplifying,
y - 3 = -3(x + 1)
y - 3 = -3x - 3
y = -3x + 3 - 3
y = -3x
Therefore, the equation of the perpendicular bisector of HI in standard form is -3x + y = 0.