FINDING EQUATION OF PERPENDICULAR BISECTOR

Write an equation for the perpendicular bisector of the line segment joining

Problem 1 :

A(1, 4) and B(6, -6)

Solution :

Step 1 :

Midpoint of AB :

Step 2 :

Slope of AB :

Step 3 :

Slope of perpendicular line = -1/m

= -1/-2

= 1/2

Step 4 :

Equation of the perpendicular bisector of AB

y - y₁ = m(x - x₁)

y + 1 = 1/2(x - 7/2)

2(y + 1) = (x - 7/2)

2y + 2 = (2x - 7)/2

2(2y + 2) = 2x - 7

4y + 4 = 2x - 7

2x - 4y = 4 + 7

2x - 4y = 11

So, equation of perpendicular bisector is 2x - 4y = 11.

Problem 2 :

A(-3, -5) and B(9, -2)

Solution :

Step 1 :

Midpoint of AB :

Step 2 :

Slope of AB :

Step 3 :

Slope of perpendicular line = -1/m

= -1/(1/4)

= -4

Step 4 :

Equation of the perpendicular bisector of AB

y - y₁ = m(x - x₁)

y + 7/2 = -4(x - 3)

(2y + 7)/2 = -4x + 12

2y + 7 = 2(-4x + 12)

2y + 7 = -8x + 24

8x + 2y = 24 - 7

8x + 2y = 17

So, equation of perpendicular bisector is 8x + 2y = 17.

Problem 3 :

A(5, 10) and B(10, 7)

Solution :

Step 1 :

Midpoint of AB :

Step 2 :

Slope of AB :

Step 2 :

Slope of the line joining the point A and B.

m = (y₂ - y₁)/(x₂ - x₁)

m = (7 - 10)/(10 - 5)

m = -3/5

Step 3 :

Slope of perpendicular line = -1/m

= -1/(-3/5)

= 5/3

Step 4 :

Equation of the perpendicular bisector of AB

y - y₁ = m(x - x₁)

y - 17/2 = 5/3(x - 15/2)

3(2y - 17)/2 = 5(x - 15/2)

(6y - 51)/2 = (5x - 75/2)

(6y - 51)/2 = (10x - 75)/2

6y - 51 = 10x - 75

10x - 6y = 75 - 51

10x - 6y = 24

So, equation of perpendicular bisector is10x - 6y = 24.

Problem 4 :

For each of the following segments :

i) Find the coordinates of the midpoint of the segment

ii) Find the gradient of the segment

iii) Find the equation of the perpendicular bisector of the segment.

Solution :

i) The endpoints are (-2, 4) and (8, 6)

Midpoint = (-2 + 8) / 2, (4 + 6) / 2

= 6/2, 10/2

= (3, 5)

ii) Find the gradient of the segment

Slope = (6 - 4) / (8 - (-2))

= 2/(8 + 2)

= 2/10

= 1/5

iii) Find the equation of the perpendicular bisector of the segment.

Slope of the perpendicular bisector = -1/(1/5)

= -5

Equation of perpendicular bisector :

(y - y₁) = m(x - x₁)

(y - 5) = -5(x - 3)

y - 5 = -5x + 15

5x + y - 5 - 15 = 0

5x + y - 20 = 0