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 = 1/3x y = 1/3x – 2 y = 3x y = 3x − 8

Answer 1

Given:

• Two coordinate points $(2,4)$ and $(4,-2)$ form a line, on a coordinate plane.
• The point $(3,1)$ is the mid- point of the line.

To find:

To select an option from the given options, that represents the line equation of the perpendicular bisector in the slope-intercept form.

Note:

• An image showing the lines is attached along with the solution.
• The point $(2,4)$ is denoted by "$A$".
• The point $(4,-2)$ is denoted by "$B$".
• The point $(3,1)$ is denoted by "$M$".
• The perpendicular bisector is denoted by "$CM$".

Formula to be used:

• The slope of the line connecting the points $A$ and $B$ should be found out to know the slope of the perpendicular bisector. It is calculated using the formula:

$m=(y_2-y_1)/(x_2-x_1)$......(1)

where, '$m$' is the indication for slope.

$(x_1,y_1)$ - point '$A$'; $(x_2,y_2)$ - point $B$

• Now, the slope of the perpendicular bisector '$CM$' is "$-(1/m)$".
• The slope- intercept form of line equation of perpendicular bisector is given by:

$(y-y_3) = (-1/m)(x-x_3)$.......(2)

where, $(x_3,y_3)$ - point $M$

Step-wise Solution:

Step-1:

• This step involves finding the slope of line '$AB$'.
• Using the equation (1), the slope line '$AB$' is calculated as follows:

$m= (y_2-y_1)/(x_2-x_1)\\m=(-2-4)/(4-2)\\m=(-6/2)\\m=-3$

⇒The slope of line $AB$ = $-3$

Step-2:

• This step involves the calculation of slope of the perpendicular bisector '$CM$'.
• Now, the slope of line $CM$ is given by:

$(-1/m)=(-1/(-3))\\(-1/m)=1/3$

⇒The slope of line $CM$ = $1/3$.

Step-3:

• This step involves the formation of line equation of the perpendicular bisector in slope-intercept form.
• The line equation in slope-intercept form is given by equation (2).

$(y-y_3)=(-1/m)(x-x_3)\\(y-1)=(1/3)(x-3)\\y-1=(1/3)x-1\\y=(1/3)x-1+1\\y=(1/3)x$

Final Solution:

The equation of the perpendicular bisector in slope-intercept form is given by $y=(1/3)x$.

Hence, the first option is correct.

Answer 2

Given : The given line segment has a midpoint at (3, 1).

line segment has end points ( 2 , 4)  and ( 4, -2 )

To Find : equation, in slope-intercept form, of the perpendicular bisector of the given line segment?

Solution:

line segment has end points ( 2 , 4)  and ( 4, -2 )

Slope  =  ( - 2 - 4)/(4 - 2)  =  - 6/2  = - 3

Slope of perpendicular line =  -1/-3  = 1/3

Equation of perpendicular bisector as it passes through ( 3 , 1)

y -  1  = (1/3) (x - 3)

=> 3y  - 3  = x  - 3

=>  y  =  x/3

y  =  x/3  is the Equation of perpendicular bisector

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