Given triangle GHI with G(4,-3), H(-4,2), and I(2,4). What is the perpendicular bisector of HI in standard form?
To find the perpendicular bisector of HI, we first need to find the midpoint of HI, which can be found by using the midpoint formula.
Midpoint formula: The coordinates of the midpoint (M) of a line segment with endpoints (x₁, y₁) and (x₂, y₂) are given by:
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Applying the midpoint formula to find the midpoint of HI:
x₁ = -4, y₁ = 2 (H)
x₂ = 2, y₂ = 4 (I)
M = ((-4 + 2)/2, (2 + 4)/2)
M = (-2/2, 6/2)
M = (-1, 3)
So, the midpoint (M) of HI is (-1, 3).
Now that we have the midpoint, we can find the slope of HI using the slope formula:
Slope formula: The slope (m) of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by:
m = (y₂ - y₁)/(x₂ - x₁)
Applying the slope formula to find the slope of HI:
x₁ = -4, y₁ = 2 (H)
x₂ = 2, y₂ = 4 (I)
m = (4 - 2)/(2 - (-4))
m = 2/6
m = 1/3
Since the perpendicular bisector of HI is perpendicular to HI, its slope will be the negative reciprocal of the slope of HI.
The negative reciprocal of 1/3 is -3.
Now that we have the midpoint and the slope of the perpendicular bisector, we can use the point-slope form of a line (y - y₁) = m(x - x₁) to find the equation of the perpendicular bisector.
Using point-slope form with the midpoint (-1, 3) and the slope -3:
(y - 3) = -3(x - (-1))
(y - 3) = -3(x + 1)
y - 3 = -3x - 3
y = -3x
The equation of the perpendicular bisector of HI in standard form is: 3x + y = 0.