Question

Write the equation in slope intercept form for the line perpendicular to c(-4,-5) and D(4,9) passing through the midpoint of the line

Answer 1

To find the equation of the line perpendicular to the line passing through points C(-4,-5) and D(4,9) and passing through the midpoint (0, 2), follow these steps: 1) Find the slope of the given line. 2) Find the midpoint of the line. 3) Find the negative reciprocal of the slope. 4) Use the slope and midpoint to write the equation of the perpendicular line in slope-intercept form.

Step-by-step explanation:

To find the equation of a line perpendicular to the line passing through points C(-4,-5) and D(4,9) and passing through the midpoint of the line, we need to follow these steps:

1. Find the slope of the given line
2. Find the midpoint of the line
3. Find the negative reciprocal of the slope to get the slope of the perpendicular line
4. Use the slope and the midpoint to write the equation of the perpendicular line in slope-intercept form

Let's go through these steps:

The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:

slope = (y₂ - y₁) / (x₂ - x₁)

In this case, the points are C(-4,-5) and D(4,9). So, we can substitute the values into the formula:

slope = (9 - (-5)) / (4 - (-4))

slope = 14 / 8

slope = 7 / 4

Step 2: Find the midpoint of the line

The midpoint of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:

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

In this case, the points are C(-4,-5) and D(4,9). So, we can substitute the values into the formula:

midpoint = ((-4 + 4) / 2, (-5 + 9) / 2)

midpoint = (0 / 2, 4 / 2)

midpoint = (0, 2)

Step 3: Find the negative reciprocal of the slope

The negative reciprocal of a slope is found by changing the sign of the slope and taking its reciprocal.

In this case, the slope is 7 / 4. So, the negative reciprocal is -4 / 7.

Step 4: Write the equation of the perpendicular line in slope-intercept form

Now that we have the slope (-4 / 7) and the midpoint (0, 2), we can use the slope-intercept form of a line to write the equation:

y = mx + b

where m is the slope and b is the y-intercept.

Substituting the values, we have:

y = (-4 / 7)x + b

To find the value of b, we can substitute the coordinates of the midpoint (0, 2) into the equation:

2 = (-4 / 7)(0) + b

2 = 0 + b

b = 2

So, the equation of the line perpendicular to the line passing through C(-4,-5) and D(4,9) and passing through the midpoint (0, 2) is:

y = (-4 / 7)x + 2

Answer 2

Slope intercept form of line passing through midpoint of CD and perpendicular to CD is
`\Rightarrow y=-(4)/(7) x+2`

Solution:

Need to find the slope intercept form for the line perpendicular to C(-4,-5) and D(4,9)

And passing through the midpoints of the line CD.

Lets first calculate slope of CD

Let say slope of CD be represented by
`m_1`

General formula of slope of line passing through points
`\left(x_(1), y_(1)\right) \text { and }\left(x_(2), y_(2)\right)` is as follows:

`m=(\left(y_(2)-y_(1)\right))/(\left(x_(2)-x_(1)\right))`

`\text { In case of line } \mathrm{CD} , x_(1)=-4, \quad y_(1)=-5 \text { and } x_(2)=4, y_(2)=9`

`\text {So slope of line } \mathrm{CD} \text { that is } m_(1)=((9-(-5)))/((4-(-4)))=(14)/(8)=(7)/(4)`

Let’s say slope of required line which is perpendicular to CD be
`m_2`

As product of slope of the lines perpendicular to each other is -1

=> slope of line CD
`*` slope of line perpendicular to CD = -1

`\begin{array}{l}{=>m_(1) * m_(2)=-1} \\\\ {\Rightarrow (7)/(4) * m_(2)=-1} \\\\ {\Rightarrow m_(2)=-(4)/(7)}\end{array}`

Now let’s find midpoint of CD

`\text { Midpoint }(x, y) \text { of two points }\left(x_(1), y_(1)\right) \text { and }\left(x_(2), y_(2)\right) \text { is given by }`

`x=(x_(2)+x_(1))/(2) \text { and } y=(y_(2)+y_(1))/(2)`

`\text { So in case of line } \mathrm{CD} , x_(1)=-4, y_(1)=-5 \text { and } x_(2)=4, y_(2)=9`

And midpoint of CD will be as follows

`x=(x_(2)+x_(1))/(2)=(4+(-4))/(2)=0 \text { and } y=(y_(2)+y_(1))/(2)=(9-5)/(2)=2`

So midpoint of CD is ( 0 , 2 )

As it is given that line whose slope intercept form is required is perpendicular to CD and passing through midpoint of CD , we need equation of line passing through ( 0 , 2 ) and having slope as
`m_(2)=-(4)/(7)`

Generic equation of line passing through
`\left(x_(1), y_(1)\right)` and having slope of m is given by

`\left(y-y_(1)\right)=m\left(x-x_(1)\right)`

`\text { In our case } x_(1)=0 \text { and } y_(1)=2 \text { and } m=-(4)/(7)`

Substituting the values in generic equation of line we get

`(y-2)=-(4)/(7)(x-0)`

As we required final equation in slope intercept form which is y = mx + c, lets rearrange our equation is required form:

`\Rightarrow y=-(4)/(7) x+2`

Hence can conclude that slope intercept form of line passing through midpoint of CD and perpendicular to CD is
`\Rightarrow y=-(4)/(7) x+2`