Question

Write the equation of the perpendicular bisector that goes through the line segment with end points of ( 10, -10 ) and B ( 2 , 2 ).

Answer 1

x - 3y = 18

Explanation:

To find the equation of the perpendicular bisector of the line segment with endpoints of (10, -10) and (2, 2), we first need to find the midpoint of the line segment. The midpoint is given by:

`\sf M =\left( ((x_1 + x_2))/(2),( (y_1 + y_2))/(2)\right)`

where
`\sf (x_1, y_1)` and
`\sf (x_2, y_2)` are the endpoints of the line segment.

In this case, the midpoint is:

`\sf M =\left( ((10+2))/(2),( (-10+2))/(2)\right) = (6,-4)`

Next, we need to find the slope of the line segment. The slope is given by:

`m = (y_2 - y_1)/(x_2 - x_1)`

where
`\sf (x_1, y_1)` and
`\sf (x_2, y_2)` are the endpoints of the line segment.

In this case, the slope is:

`\sf m = (2 - (-10))/(2 - 10) = (12)/(-8) = - (3)/(2)`

The slope of the perpendicular bisector is the negative reciprocal of the slope of the line segment. Therefore, the slope of the perpendicular bisector is:

`\sf m = -\left( - (2)/(3)\right)`

`\sf m = (2)/(3)`

Now that we know the midpoint and slope of the perpendicular bisector, we can use the point-slope form of linear equations to find the equation of the perpendicular bisector.

The point-slope form is given by:

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

where
`\sf (x_1, y_1)` is a point on the line and m is the slope of the line. In this case, we know that the midpoint (6, -4) is on the perpendicular bisector and the slope of the perpendicular bisector is 2/3. Therefore, the equation of the perpendicular bisector is:

`\sf y - (-4) = (2)/(3) (x - 6)`

Simplifying the equation, we get:

`\sf y + 4 = (2)/(3) (x - 6)`

3y + 12 = x - 6

3y = x - 6 - 12

3y = x - 18

x - 3y -18 = 0

x - 3y = 18

Therefore, the equation of the perpendicular bisector is:

x - 3y = 18