Q:

# A line segment has endpoints at (4, –6) and (0, 2). What is the slope of the given line segment? What is the midpoint of the given line segment? What is the slope of the perpendicular bisector of the given line segment? What is the equation, in slope-intercept form, of the perpendicular bisector?

Accepted Solution

A:
1) slope = (y₂-y₁)/(x₂-x₁)

Let A and B be A(4,-6) and B(0,2) ;

m = [2-(-6)]/[0-4) = (2+6)/(-4) → m = -2

2) Midpoint = value of x of the midpoint = (x₁+x₂)/2
value of y of the midpoint = (y₁+y₂)/2

x(midpoint) =  (4+0)/2  → x= 2
y(midpoint) =  (-6+2)/2 → y= - 2, so Midpoint M(2,-2)

3) Slope of the perpendicular bisector  to AB:
The slope of AB = m = -2
Any perpendicular to AB will have a slope m' so that m*m' = -1 (or in other term, the slope of one is inverse reciprocal of the second, then if m =-2, then m' = +1/2 ; Proof [ (-2)(1/2) = -1]

4) Note that the perpendicular bisector of AB passes through the midpoint of AB or M(2,-2). Moreover we know that the slope of the bisector is m'= 1/2
The equation of the linear function is :

y = m'x + b or y = (1/2)x + b. To calculate b, replace x and y by their respective values [in M(-2,2)]

2= (1/2).(-2) + b → 2 = -1 + b → and b= 3, hence the equation is:

y = (1/2)x + 3