To understand the term perpendicular bisector, you need to break it down:
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Team Equation of a Perpendicular Bisector Teachers
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:
Fig. 1 - A 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.
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?
Identify the original gradient: In the equation y = mx + c, m is the gradient. Therefore, the gradient of the original line is 3.
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?
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.
A line segment is a part of a line between two points.
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: \(\left(\frac{a+c}{2}, \frac{b+d}{2}\right)\).
Fig. 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.
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).
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)
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?
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?
Therefore, the equation for the perpendicular bisector of the line segment is
\(y - \frac{21}{2} = - \frac{9}{7}(x - \frac{21}{2})\) \(y - \frac{21}{2} = - \frac{9}{7}x + \frac{189}{14}\frac{9}{7}x + y = \frac{189}{14} + \frac{21}{2}\frac{9}{7}x + y = \frac{189+147}{14}\frac{9}{7}x + y = \frac{189+147}{14}\frac{9}{7}x + y = \frac{336}{14}\frac{9}{7}x + y = 24\frac{9}{7}x + y - 24 = 0\)
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.
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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:
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