Question

The given line segment has a midpoint at (3, 1).On a coordinate plane, a line goes through (2, 4), (3, 1), and (4, negative 2).What is the equation, in slope-intercept form, of the perpendicular bisector of the given line segment?y = One-thirdxy = One-thirdx – 2y = 3xy = 3x − 8

Answer 1

Given that:

- The line segment has a midpoint at:

$(3,1)$

- The line goes through these points:

$(2,4),(3,1),(4,-2)$

You need to remember that the equation of a line in Slope-Intercept Form is:

$y=mx+b$

Where "m" is the slope of the line and "b" is the y-intercept.

You can find the slope of the line that passes through the points shown above, by using this formula:

$m=\frac{y_2-y_1}{x_2-x_1}$

Where these two points are on the line:

$(x_1,y_1),(x_2,y_2)$

You can substitute the coordinates of two of the three given points on the line. You can use these points:

$(2,4),(4,-2)$

Set up that:

$\begin{gathered} y_2=-2 \\ y_1=4 \\ x_2=4 \\ x_1=2 \end{gathered}$

Then, you get:

$m=\frac{-2-4}{4-2}=\frac{-6}{2}=-3$

By definition, the slopes of perpendicular lines are opposite reciprocals. Therefore, you can determine that the slope of the perpendicular bisector of the given line segment is:

$m_{bisector}=\frac{1}{3}$

Substitute the slope and the coordinates of the Midpoint into the following equation, and then solve for "b":

$y=(m_{bisector)}(x)+b$

Then, you get:

$\begin{gathered} 1=(\frac{1}{3})(3)+b \\ \\ 1=1+b \\ 1-1=b \\ b=0 \end{gathered}$

Knowing the slope and the y-intercept, you can write this equation in Slope-Intercept Form:

$y=\frac{1}{3}x+0$

Simplify:

$y=\frac{1}{3}x$

Hence, the answer is: First option.