# The Perpendicular Bisector

Hi. The topic here is on the perpendicular bisector found in high school mathematics (in Ontario, Canada). It is assumed that the reader knows the slope of a line between two points, the midpoint between two points on a line and the point-slope equation form of a line.

What Is A Perpendicular Bisector?

Examples

Practice Problems

Solutions

What Is A Perpendicular Bisector?

Suppose we have a point with co-ordinates (, ) and a point with co-ordinates (, ). A line can be connected from point to point . The slope of this line can easily computed using . The midpoint between points and can be computed using (, ).

After finding the slope and midpoint of this line , we can construct the perpendicular bisector. This perpendicular bisector line goes through the line at a 90 degree angle and through the midpoint (, ) as well.

The slope of this perpendicular bisector is the negative reciprocal of the slope of the line . That is, the slope of the perpendicular bisector is .

Recall that horizontal lines have a slope of zero and vertical lines have undefined (infinite) slopes. A horizontal line is perpendicular to a vertical line. The converse of the previous statement is true as well. That is, a vertical line is perpendicular to a horizontal line.

Examples

Example One

If the slope of line is then the slope of the perpendicular bisector line which goes through the midpoint of line at a 90 degree angle is .

Example Two

Suppose that the slope of the line is then the slope of the perpendicular bisector line which goes through the midpoint of line at a 90 degree angle is .

Example Three

Here is a more involved example.

We have a point with the co-ordinate and a point at (3, -1). Find the slope of the perpendicular bisector which gros through the midpoint of the line . In addition, determine the equation of this perpendicular bisector line passing through this midpoint.

Solution:

The slope of the line is calculated first.

Denote the midpoint of as . This midpoint is computed as follows:

The slope of this perpendicular bisector line is the negative reciprocal of . This negative reciprocal would be .

Recall that the point-slope form of the equation of a line with a point () is of the form:

We substitute the slope of the perpendicular bisector line which is and the midpoint as our (). After substitution, we isolate for to obtain the form.

Source: http://quicklatex.com/cache3/68/ql_b048b8726b1050bab48eac82ace46868_l3.png

Practice Problems

1) Suppose that the slope of the line is . What is the slope of the perpendicular bisector line which goes through the midpoint of line at a 90 degree angle?

2) Suppose that the slope of the line is . What is the slope of the perpendicular bisector line which goes through the midpoint of line at a 90 degree angle?

3) You are given two points G (-2, 1) and H (0, 7). What is the equation of the perpendicular bisector line which passes through the midpoint of the line GH?

Solutions

1)

2)

3) Midpoint of line GH is . The slope of line GH is . The negative reciprocal of is . The equation of the perpendicular bisector line is .