Question

Two points A and B have coordinates (1, 3) and (9, −1), respectively. The perpendicular bisector of  AB intersects the y-axis at the point C. Find the coordinates of C.

Solution

Coordinates of point C which is y-intercept of the perpendicular bisector of AB.

The point at which curve (or line) intercepts y-axis, the value of . So we can find the  value of coordinate by substituting in the equation of the curve (or line). 

Therefore, we need equation of the perpendicular bisector of AB. 

To find the equation of the line either we need coordinates of the two points on the line (Two-Point  form of Equation of Line) or coordinates of one point on the line and slope of the line (Point-Slope form of Equation of Line).

We have neither coordinate of a point on the perpendicular bisector of AB nor its slope.

First, we find coordinates of a point on the perpendicular bisector of AB.

Since, we are looking for coordinates of a point on the perpendicular bisector of AB, it can be mid- point of AB.

Let’s find coordinates of mid-point of AB.

To find the mid-point of a line we must have the coordinates of the end-points of the line.

Expressions for coordinates of mid-point of a line joining points  and;

x-coordinate of mid-point of the line

y-coordinate of mid-point of the line

We are given that points A and B have coordinates (1, 3) and (9, −1), respectively.

Therefore; 

x-coordinate of mid-point  of the line

y-coordinate of mid-point  of the line

Hence, coordinates of a point on the perpendicular bisector of AB, the mid-point of AB, are M(5, 1).

Next, we need slope of the perpendicular bisector of AB.

If two lines are perpendicular (normal) to each other, then product of their slopes and  is;

Therefore, if we have slope of the line AB, we can find the slope of the perpendicular bisector of AB.

We need to find slope of the line AB.

Expression for slope of a line joining points and ;

Therefore, for coordinates of points A and B (1, 3) and (9, −1), respectively;

Hence;

Now, with coordinates of a point M(5, 1) and slope of the line , we can write equation of the  perpendicular bisector of AB.

Point-Slope form of the equation of the line is;

Next, we need to find y-coordinate of point C.

We substitute ;

Hence, coordinates of the point C are (0, -9).