Question

Find an equation for the perpendicular bisector of the line segment whose endpoints are (-9,1) and (3,5).

Answer 1

the equation of the perpendicular bisector of the line segment with endpoints (-9, 1) and (3, 5) is y = -3x - 6.

Explanation:

To find the equation of the perpendicular bisector of a line segment, we need to find the midpoint of the line segment and determine the slope of the line segment. Let's go step by step.

Step 1: Find the midpoint of the line segment.

The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is given by the formula:

Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)

Using the endpoints (-9, 1) and (3, 5):

Midpoint = ((-9 + 3) / 2, (1 + 5) / 2)

= (-6 / 2, 6 / 2)

= (-3, 3)

So, the midpoint of the line segment is (-3, 3).

Step 2: Find the slope of the line segment.

The slope of a line segment with endpoints (x1, y1) and (x2, y2) is given by the formula:

Slope = (y2 - y1) / (x2 - x1)

Using the endpoints (-9, 1) and (3, 5):

Slope = (5 - 1) / (3 - (-9))

= 4 / 12

= 1/3

So, the slope of the line segment is 1/3.

Step 3: Find the negative reciprocal of the slope.

The negative reciprocal of a slope is obtained by flipping the fraction and changing its sign.

The negative reciprocal of 1/3 is -3/1 or -3.

Step 4: Write the equation of the perpendicular bisector.

The equation of a line with slope m and passing through the point (x1, y1) is given by the point-slope form:

y - y1 = m(x - x1)

Using the midpoint (-3, 3) and the negative reciprocal slope -3:

y - 3 = -3(x - (-3))

y - 3 = -3(x + 3)

y - 3 = -3x - 9

y = -3x - 9 + 3

y = -3x - 6

Answer 2

`y=-3x-6`

Explanation:

The perpendicular bisector of a line segment is a line that divides the line segment into two equal parts and is perpendicular (at a right angle) to the line segment.

The perpendicular bisector intersects the line segment at its midpoint, so we first need to find the midpoint of the line segment.

`\boxed{\begin{minipage}{7.4 cm}\underline{Midpoint between two points}\\\\Midpoint $=\left((x_2+x_1)/(2),(y_2+y_1)/(2)\right)$\\\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ are the endpoints.\\\end{minipage}}`

Given the endpoints of the line segment are (-9, 1) and (3, 5), substitute these into the midpoint formula to find the midpoint of the line segment:

`\textsf{Midpoint}=\left((3+(-9))/(2),(5+1)/(2)\right)=\left(-3,3\right)`

Now we need to determine the slope of the line segment by substituting the endpoints into the slope formula:

`\textsf{slope}=(y_2-y_1)/(x_2-x_1)=(5-1)/(3-(-9))=(4)/(12)=(1)/(3)`

Take the negative reciprocal of the line segment slope to obtain the slope of the perpendicular bisector:

`\textsf{Slope of perpendicular bisector}= -3`

Finally, to find the equation of the perpendicular bisector, substitute the found slope and the midpoint into the point-slope formula:

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

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

`y-3=-3(x+3)`

`y-3=-3x-9`

`y=-3x-6`

Therefore, the equation for the perpendicular bisector of the line segment whose endpoints are (-9, 1) and (3, 5) is:

`\boxed{y=-3x-6}`