Question

If M is the midpoint of segment AB, find the coodinates of B; given that A = (-8,9) and M= (-6,6).  Calulate the distance between A and B

Find the equation in point-slope form of the line that is the perpendicular bisector of the segment between (16,-4) and (-2,-76).

Answer 1

1) Mid point of segment AB = M

A(x₁ , y₁) = (-8, 9) B(x₂, y₂) = (x, y)

Mid point between the points = [(x₁+ x₂)/2 , (y₁ + y₂)/2]

                                       (-6,6)  = [(- 8 + x)/2 , (9 + y)/2]

(-8 + x)/2 = -6 and (9 + y)/2 = 6

-8 + x = - 12 and 9 + y = 12

x = - 4 and y = 3

Coordinates of B(- 4, 3)

Let the points (x₁ , y₁) = (-8, 9) and (x₂, y₂) = (- 4, 3)

Distance between the points A and B= √[(x₂ - x₁)² + (y₂ - y₁)²]

                                                = √[(-4 + 8)² + (3 - 9)²]

                                                = √(16 + 36)

 AB = √52

2) End points of the segment are (16, -4) and (-2, -76).

Slope - intercept form line equation is y = mx + b, where m is slope and b is y - intercept.

Let the points are (x₁, y₁) =(16, -4) and (x₂, y₂) = (-2, -76).

Slope (m) = [(y₂ - y₁)/(x₂ -x₁)]

m = [(- 76 + 4)/(- 2 - 16)]

m  = [- 72/-18]

⇒ m = 4.

Because the slopes of perpendicular lines are negative reciprocals, the slope of perpendicular is -1/4.

Now, the perpendecular line equation is y = -x/4 + b.

Find the y - intercept, we need a point on the perpendicular bisector, to find this point we will find the midpoint of the two ponts:

midpoint = [ (16 - 2)/2, (- 4 -76)/2 ]

               = [ 7, -40 ]

            

Find the y - intercept by substituting the midpoint in the perpendecular line equation say (x, y) = (7, -40).

-40 = (-7/4) + b

b = -40 + 7/4

⇒ b = - 153/4

The perpendicular line equation  is y = -x/4 - 153/4.