# FIND EQUATION OF PERPENDICULAR BISECTOR WITH TWO POINTS

A perpendicular bisector of a line segment is a line segment perpendicular to and passing through the midpoint of the line.

To find equation of perpendicular bisector, we follow the steps given below.

Step 1 :

Find the midpoint of the line segment for which we have to find the perpendicular bisector.

Step 2 :

Find the slope of the line segment.

Step 3 :

Find the slope of the perpendicular line using the formula -1/m. Here m is slope of the given line.

Step 4 :

To find equation of the perpendicular line, we use the formula given below.

(y - y1) = -1/m (x - x1)

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

Problem 1 :

A(-3, 4) and B(5, 6)

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 - 5 = -4(x - 1)

y = -4x + 4 + 5

y = -4x + 9

So, equation of perpendicular bisector is y = -4x + 9.

Problem 2 :

A(3, 8) and B(7, 14)

Solution :

Step 1 :

Midpoint of AB :

Step 2 :

Slope of AB :

Step 3 :

Slope of perpendicular line = -1/m

= -1/(3/2)

= -2/3

Step 4 :

Equation of the perpendicular bisector of AB

y - y1 = m(x - x1)

y - 11 = -2/3(x - 5)

3(y - 11) = -2(x - 5)

3y - 33 = -2x + 10

2x + 3y = 10 + 33

2x + 3y = 43

So, equation of perpendicular bisector is 2x + 3y = 43.

Problem 3 :

A(-5, 6) and B(1, 8)

Solution :

Step 1 :

Midpoint of AB :

Step 2 :

Slope of AB :

Step 3 :

Slope of perpendicular line = -1/m

= -1/(1/3)

= -3

Step 4 :

Equation of the perpendicular bisector of AB

y - y1 = m(x - x1)

y - 7 = -3(x + 2)

y - 7 = -3x - 6

3x + y = 7 - 6

3x + y = 1

y = -3x + 1

So, equation of perpendicular bisector is y = -3x + 1.

Problem 4 :

A(-3, -6) and B(-1, 2)

Solution :

Step 1 :

Midpoint of AB :

Step 2 :

Slope of AB :

Step 3 :

Slope of perpendicular line = -1/m

= -1/4

Step 4 :

Equation of the perpendicular bisector of AB

y - y1 = m(x - x1)

y + 2 = -1/4(x + 2)

4(y + 2) = -1(x + 2)

4y + 8 = -x - 2

x + 4y = -8 - 2

x + 4y = -10

So, equation of perpendicular bisector is x + 4y = -10.

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