Given triangle GHI with G(4,-3), H(-4,2), and I(2,4). What is the perpendicular bisector of HI in standard form?

Answer 1

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.