The equation y = (13 / 4) · x - 19 / 2 represents the perpendicular bisector of the line segment.
<h3>How to find the equation for the perpendicular bisector of a line segment</h3>
In this problem we need to determine the equation of the line perpendicular to a line segment whose endpoints are known. The equation of the line is represented by a first grade polynomial of the form:
y = m · x + b
Where:
- m - Slope
- b - Intercept
- x - Independent variable.
- y - Dependent variable.
And the relationship between the slopes of two perpendicular lines:
m · m' = - 1
Where:
- m - Slope of the original line.
- m' - Slope of the perpendicular line.
And the slope of the original line can be found by the secant line formula:
m = Δy / Δx
And the midpoint of a line segment between two endpoints is defined below, necessary for the location of the bisector:
M(x, y) = 0.5 · A(x, y) + B(x, y)
Where:
- M(x, y) - Midpoint
- A(x, y), B(x, y) - Endpoints
First, find the slope of the line segment by secant line formula:
m = [- 1 - (- 5)] / (- 5 - 9)
m = - 4 / 13
Second, find the slope of the perpendicular bisector by the slope relationship between two perpendicular lines:
m' = - 1 / m
m' = - 1 / (- 4 / 13)
m' = 13 / 4
Third, determine the midpoint of the line segment by the midpoint formula:
M(x, y) = 0.5 · (9, - 5) + 0.5 · (- 5, - 1)
M(x, y) = (4.5, - 2.5) + (- 2.5, - 0.5)
M(x, y) = (2, - 3)
Fourth, find the intercept of the perpendicular bisector by means of the equation of the line:
b = y - m' · x
b = - 3 - (13 / 4) · 2
b = - 3 - 26 / 4
b = - 3 - 13 / 2
b = - 19 / 2
The equation for the perpendicular bisector is y = (13 / 4) · x - 19 / 2.
To learn more on perpendicular bisectors: brainly.com/question/11900712
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