Equation of a Perpendicular Bisector

To understand the term perpendicular bisector, you need to break it down:

Equation of a Perpendicular Bisector Equation of a Perpendicular Bisector

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Contents
Table of contents
    • Perpendicular: lines that meet at a right angle (90°)

    • Bisector: the partition of a line into two equal parts

    Therefore, a perpendicular bisector is when a line is partitioned at a right angle by another line into two equal parts- as seen below:

    Equation of a Perpendicular Bisector Diagram of a Perpendicular Bisector StudySmarterFig. 1 - A perpendicular bisector.

    Finding the equation for the perpendicular bisector

    A perpendicular bisector is expressed as a linear equation. To create an equation for the perpendicular bisector of a line, you first need to find the gradient of the slope of the perpendicular bisector and then substitute the known coordinates into a formula: either, \(y = m \cdot x + c\) or \(y - y_1 = m (x - x_1)\). If the coordinate of the bisection is not known, you will need to find the midpoint of the line segment.

    Find the gradient of the slope of the perpendicular bisector

    • The first step of creating an equation for the perpendicular bisector is to find the gradient of its slope. Because the slopes of the original line and the bisector are perpendicular, we can use the gradient of the original line to work out the gradient of the perpendicular bisector.

    • The gradient of the perpendicular bisector is the inverse reciprocal of the slope of the original line. The gradient of the perpendicular bisector can be expressed as -1 / m, where m is the gradient of the slope of the original line.

    Line a has the equation \(y = 3x + 6\), is perpendicularly bisected by the line l. What is the gradient of line a?

    1. Identify the original gradient: In the equation y = mx + c, m is the gradient. Therefore, the gradient of the original line is 3.

    2. Find the gradient of the slope of the perpendicular bisector: Substitute the original gradient, 3, into the formula \(-\frac{1}{m}\) to find the inverse reciprocal because it is perpendicular. Therefore, the gradient of the line is \(-\frac{1}{3}\).

    If you are not given the original equation, you might first have to work out the gradient of the equation of the line using two coordinates. The formula for the gradient is \(\frac{{y_2 - y_1}}{{x_2 - x_1}}\).

    Line 1 stems from (3, 3) to (9, -21) and is perpendicularly bisected by Line 2. What is the gradient of the slope of Line 2?

    1. Identify the original gradient: As we don't have the equation for line 1 we'll need to calculate the gradient of its slope. To find the gradient of Line 1, you'll need to substitute the coordinates into the gradient formula: \( gradient = \frac{change \, in \, y}{change \, in \, x} \). Therefore, \(\frac{-21 - 3}{9 - 3} = \frac{-24}{6} = -4\).
    2. Find the gradient of the perpendicular bisector: Substitute -4 into the formula \(-\frac{1}{m}\), because the lines are perpendicular. Therefore, the gradient is \(\frac{-1}{-4}\), which is equal to \(\frac{1}{4}\).

    Finding the midpoint of a line segment

    The midpoint is a coordinate that shows the halfway point of a line segment. If you are not given the equation of the original line, you will have to calculate the midpoint of the line segment as this is where the bisector will intersect with the original line.

    You can find the midpoint by averaging from the x and y coordinates of the line segment end. For instance, you can find the midpoint of the segment of the line with the endpoints (a, b) and (c, d) through the formula: \(\left(\frac{a+c}{2}, \frac{b+d}{2}\right)\).

    Equation of a Perpendicular Bisector Perpendicular Diagram graph StudySmarterFig. 2 - A perpendicular bisector on a graph

    A segment of a line has the endpoints (-1, 8) and (15, 10). Find the coordinates of the midpoint.

    • Using \(\left(\frac{a+c}{2},\frac{b+d}{2}\right)\), substitute in the endpoints (-1, 8) and (15, 10) to get \(\left(-\frac{1+15}{2}, \frac{8+10}{2}\right)\)= (7, 9)

    You can rearrange the formula to use the midpoint to find one of the other coordinates.

    AB is a segment of a line with a midpoint of (6, 6). Find B when A is (10, 0).

    • You can partition \(\left(\frac{a+c}{2}, \frac{b+d}{2}\right)\)into parts relating to the x- and y- coordinate where the centre is (m, n)
      • X coordinate: \(\frac{a + c}{2}\)= m
      • Y coordinates: \(\frac{b+d}{2} = n\)
    • Then, you can substitute the known coordinates into these new Equations

      • X coordinates: \(\frac{10+c}{2} = 6\)

      • Y coordinates:\(\frac{0+d}{2}=6\)

    • Rearranging these Equations would give you c = 2 and d = 12. Therefore, B = (2, 12)

    Creating the equation of a perpendicular bisector

    To finish formulating the equation for the perpendicular bisector, you need to substitute the gradient of the slope as well as the point of bisection (the midpoint) into a linear equation formula.

    These formulas include:

    \(y = m \cdot x + c\)

    \(y - y_1 = m(x - x_1)\)

    \(Ax + By = C\)

    You can substitute directly into the first two formulas whilst the last one needs to be rearranged into that form.

    A segment of a line from (4,10) to (10, 20) is perpendicularly bisected by line 1. What is the equation of the perpendicular bisector?

    1. Find the gradient of the slope of the original line: \(\frac{20 - 10}{10 - 4} = \frac{10}{6} = \frac{5}{3}\)
    2. Find the gradient of the slope of line 1: \(-\frac{1}{m} = -\frac{1}{\frac{5}{3}} = -\frac{3}{5}\)
    3. Find the midpoint of the line segment: \(\left(\frac{4+10}{2}, \frac{10+20}{2}\right) = (7, 15)\)
    4. Substitute into a formula: \(y - 15 = -\frac{3}{5}(x - 7)\)
    Therefore, the equation for the perpendicular bisector of the line segment is\(y - 15 = -\frac{3}{5}(x - 7)5y - 75 = -3x + 21 + 3x + 5y - 96 = 0\)

    A segment of a line from (-3, 7) to (6, 14) is perpendicularly bisected by line 1. What is the equation of the perpendicular bisector?

    1. Find the gradient of the slope of the original line: \(\frac{14-7}{6 - (-3)} = \frac{7}{9}\)
    2. Find the gradient of the slope of line 1: \(-\frac{1}{m} = -\frac{1}{\frac{7}{9}} = -\frac{9}{7}\)
    3. Find the midpoint of the line segment: \(\left(-\frac{3}{2}+6, \frac{7}{2}+14 \right) = \left(\frac{3}{2}, \frac{21}{2}\right)\)
    4. Substitute into a formula: \(y - \frac{21}{2} = -\frac{9}{7}(x - \frac{21}{2})\)

    Therefore, the equation for the perpendicular bisector of the line segment is

    \(y - \frac{21}{2} = - \frac{9}{7}(x - \frac{21}{2})\) \(y - \frac{21}{2} = - \frac{9}{7}x + \frac{189}{14}\frac{9}{7}x + y = \frac{189}{14} + \frac{21}{2}\frac{9}{7}x + y = \frac{189+147}{14}\frac{9}{7}x + y = \frac{189+147}{14}\frac{9}{7}x + y = \frac{336}{14}\frac{9}{7}x + y = 24\frac{9}{7}x + y - 24 = 0\)

    Equation of a Perpendicular Bisector - Key takeaways

    • A perpendicular bisector is a line that perpendicularly splits another line in half. The perpendicular bisector is always expressed as a linear equation.

    • To calculate the gradient of a perpendicular line, you take the negative reciprocal of the gradient of the slope of the original line.

    • If you are not given an equation for the slope of the original line, you need to find the midpoint of the segment as this is the point of bisection. To calculate the midpoint, you substitute the endpoints of a line segment into the formula:\(\left(\frac{a+c}{2},\frac{b+d}{2}\right)\)

    • To create the equation for the perpendicular bisector, you need to substitute the midpoint and the gradient into a linear equation formula.

    Frequently Asked Questions about Equation of a Perpendicular Bisector

    What is the perpendicular bisector of a line?

    A perpendicular bisector is a line that perpendicularly (at an angle 90) splits another line in half

    What is the equation of a perpendicular bisector?

    The equation of a perpendicular bisector is a linear equation which tells the line which splits another line in half perpendicularly.

    How do you find the perpendicular bisector of two points?

    To create an equation of perpendicular bisector:

    1. First, you need to find the gradient of the slope original line by substituting the endpoints into the formula: change in y/ change in x
    2. Then, you find the negative reciprocal of the original gradient by substituting it into -1/m, where m is the gradient of the slope of the original line. If necessary, you then find the midpoint of the line segment (a,b) to (c,d) by averaging the x and y values. 
    3. You then create the equation of the perpendicular bisector by substituting the midpoint and the gradient into an equation formula.

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