# What is the equation of the perpendicular bisector of a chord of a circle?

##### 2 Answers

For a chord AB, with

#### Explanation:

Supposing a chord AB with

The midpoint is

The slope of the segment defined by A and B (the chord) is

The slope of the line perpendicular to the segment AB is

The equation of the line required is

#y-y_M=p(x-x_M)#

#y-(y_A+y_B)/2=-(x_B-x_A)/(y_B-y_A)*(x-(x_A+x_B)/2)#

Or

#y=-(x_B-x_A)/(y_B-y_A)*(x-(x_A+x_B)/2)+(y_A+y_B)/2#

Note: the center of the circle, assummedly point C

The equation of the perpendicular bisector of a chord of a circle is the equation of a diameter of the circle.

#### Explanation:

Let the equation of the circle be standard one having center at origin and radius **r**

The coordinates of the end points of the chord AB

The coordinate of the middle point C (x',y') of AB

The slope of AB

Slope of the perpendicular bisector of AB is

The equation of the perpendicular bisector of AB

OR

Obviously this is the equation of a straight line passing through the origin (0,0),the center of the circle. So the perpendicular bisector of the chord is a diameter of the circle.