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=(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=(-2-4)/(4-2)=(-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)=(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=((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=(1)/(3)x+0`

Simplify:

`y=(1)/(3)x`

Hence, the answer is: First option.