Question

How do you find the perpendicular bisector of a line segment for the given points (2, 5) and (8, 3) ?

Answer 1

Hint: To find the perpendicular bisector of a line segment all you need to do is find their midpoint and negative reciprocal, and plug these answers into the equation for a line in slope-intercept form.

Formula used:
\[y = mx + b\]
y is equation of a line
m is slope.
m =\[slope = \dfrac{{y_2 - y_1}}{{x_2 - x_1}}\]
Midpoint = \[\left( {\dfrac{{x_1 + x_2}}{2},\dfrac{{y_1 + y_2}}{2}} \right)\]
$(x_1, y_1)$ are the coordinates of x
$(x_2, y_2)$ are the coordinates of y

Complete step-by-step solution:
To explain this let us consider the perpendicular bisector of a line segment consisting of two points.
Considering the points (2, 5) and (8, 3).
First, we need to find out the mid points using the formula
Midpoint = \[\left( {\dfrac{{x_1 + x_2}}{2},\dfrac{{y_1 + y_2}}{2}} \right)\]
$(x_1, y_1)$ are the coordinates of (2, 5)
$(x_2, y_2)$ are the coordinates of (8, 3)
Now substitute the values, we get
\[\Rightarrow \left( {\dfrac{{2 + 8}}{2},\dfrac{{5 + 3}}{2}} \right)\]
\[\Rightarrow \left( {\dfrac{{10}}{2},\dfrac{8}{2}} \right)\]
\[\Rightarrow \left( {5,4} \right)\]
Hence, the coordinates of the midpoint of (2, 5) and (8, 3) are \[\left( {5,4} \right)\]
Next, we need to find the slope of a line in which the slope of a line measures the distance of its vertical change over the distance of its horizontal change.
\[\Rightarrow slope = \dfrac{{y_2 - y_1}}{{x_2 - x_1}}\]
\[\Rightarrow \dfrac{{3 - 5}}{{8 - 2}}\]
\[\Rightarrow \dfrac{{ - 2}}{6}\]
Therefore,
\[\Rightarrow slope = \dfrac{{ - 1}}{3}\]
Find the negative reciprocal of the slope of the two points i.e.,
\[\Rightarrow slope{^{ - 1}} = 3\]= m
Calculate the equation of a line, which is given by
\[\Rightarrow y = mx + b\]
\[\Rightarrow y = 3x + b\]
Since the perpendicular bisector runs through the midpoint of the two lines, you can plug the coordinates of the midpoint into the equation of the line. Simply plug in (5, 4) into the x and y coordinates of the line.
\[\Rightarrow y = 3x + b\]
\[\Rightarrow 4 = 3\left( 5 \right) + b\]
\[\Rightarrow 4 = 15 + b\]
To solve for the remaining variable, "b," which is the y-intercept of this line. Isolate the variable "b" to find its value.
Subtract 15 from both sides of the equation.
\[\Rightarrow 4 - 15 = b\]
\[\Rightarrow - 11 = b\]
$\Rightarrow b = -11.$
Therefore, equation of perpendicular bisector of the points (2, 5) and (8, 3) is
\[\Rightarrow y = mx + b\]
\[\Rightarrow y = 3x - 11\]

Note: When the equation of a line using the slope of the line and a point through which the line passes, that equation can be found using the point-slope formula. The equation of a line whose slope is m and which passes through a point $(x_1, y_1)$ is found using the point-slope form and is given as \[y - y_1 = m\left( {x - x_1} \right)\].