Question

Q28
Please help me as soon as possible.

Answer 1

`\textsf{F.} \quad y=(1)/(3)x`

Explanation:

A perpendicular bisector is a line that intersects another line segment at 90°, dividing it into two equal parts.

To determine the equation of the perpendicular bisector of the segment with endpoints (2, 4) and (4, -2), we first need to find the slope of the line segment and the midpoint of the line segment.

To find the slope of the line segment with endpoints (2, 4) and (4, -2), substitute the endpoints into the slope formula:

`\textsf{Slope}\:(m)=(y_2-y_1)/(x_2-x_1)=(-2-4)/(4-2)=(-6)/(2)=-3`

If two lines are perpendicular to each other, their slopes are negative reciprocals. Therefore, the slope of the perpendicular bisector is:

`\textsf{Slope\;of\;perpendicular bisector:}\;\;m=(1)/(3)`

To find the midpoint of the line segment with endpoints (2, 4) and (4, -2), substitute the endpoints into the midpoint formula:

`\textsf{Midpoint}=\left((x_2+x_1)/(2),(y_2+y_1)/(2)\right)`

`\textsf{Midpoint}=\left((4+2)/(2),(-2+4)/(2)\right)`

`\textsf{Midpoint}=\left((6)/(2),(2)/(2)\right)`

`\textsf{Midpoint}=\left(3,1\right)`

To find the equation of the perpendicular bisector, substitute the found slope m = 1/3 and point (3, 1) into the point-slope form of a linear equation:

`y-y_1=m(x-x_1)`

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

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

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

Therefore, the equation of the perpendicular bisector of the segment with endpoints (2, 4) and (4, -2) is:

`\Large\boxed{\boxed{y=(1)/(3)x}}`