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.


Table Of Contents

What Is A Perpendicular Bisector?

Examples

Practice Problems

Solutions


What Is A Perpendicular Bisector?

Suppose we have a point A with co-ordinates (x_1, y_1) and a point B with co-ordinates (x_2, y_2). A line AB can be connected from point A to point B. The slope of this line can easily computed using m_{AB} = \dfrac{y_2 - y_1}{x_2 - x_1}. The midpoint between points A and B can be computed using (\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}).

After finding the slope and midpoint of this line AB, we can construct the perpendicular bisector. This perpendicular bisector line goes through the line AB at a 90 degree angle and through the midpoint (\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}) as well.

The slope of this perpendicular bisector is the negative reciprocal of the slope of the line AB. That is, the slope of the perpendicular bisector is -\dfrac{1}{m_{AB}}.

 

Source: http://math-problems.math4teaching.com/wp-content/uploads/2013/11/perpendicular_bisector.png

 

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 CD is m_{CD} = 3 then the slope of the perpendicular bisector line which goes through the midpoint of line CD at a 90 degree angle is -\dfrac{1}{m_{CD}} = -\dfrac{1}{3}.

Example Two

Suppose that the slope of the line JK is m_{JK} = \dfrac{-2}{5} then the slope of the perpendicular bisector line which goes through the midpoint of line JK at a 90 degree angle is -\dfrac{1}{m_{JK}} = -\dfrac{1}{-\dfrac{2}{5}} = \dfrac{5}{2}.

Example Three

Here is a more involved example.

We have a point E with the co-ordinate (1, 5) and a point F at (3, -1). Find the slope of the perpendicular bisector which gros through the midpoint of the line EF. In addition, determine the equation of this perpendicular bisector line passing through this midpoint.

Solution:

The slope of the line EF is calculated first.

    \[m_{EF} = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{-1 - 5}{3 - 1} = \dfrac{-6}{2} = -3\]

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

    \[G = (\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}) = (\dfrac{1 + 3}{2}, \dfrac{5 - 1}{2}) = (\dfrac{4}{2}, \dfrac{4}{2}) = (2, 2)\]

The slope of this perpendicular bisector line is the negative reciprocal of m_{EF} = - 3. This negative reciprocal would be \dfrac{1}{3}.

Recall that the point-slope form of the equation of a line with a point (x_{o}, y_{o}) is of the form:

    \[(y - y_{o}) = m(x - x_{o})\]

We substitute the slope of the perpendicular bisector line which is \dfrac{1}{3} and the midpoint (2, 2) as our (x_{o}, y_{o}). After substitution, we isolate for y to obtain the y = mx + b form.

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

 


Practice Problems

1) Suppose that the slope of the line PQ is m_{PQ} = -2. What is the slope of the perpendicular bisector line which goes through the midpoint of line PQ at a 90 degree angle?

2) Suppose that the slope of the line XY is m_{XY} = -\dfrac{2}{3}. What is the slope of the perpendicular bisector line which goes through the midpoint of line XY 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) \dfrac{1}{2}

2) \dfrac{3}{2}

3) Midpoint of line GH is (-1 , 4). The slope of line GH is m_{GH} = 3. The negative reciprocal of 3 is -\dfrac{1}{3}. The equation of the perpendicular bisector line is y = -\dfrac{1}{3}x + \dfrac{11}{3}.

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