To find the equation of the line [tex]\(\overleftrightarrow{BC}\)[/tex] given that [tex]\(\overleftrightarrow{AB}\)[/tex] and [tex]\(\overleftrightarrow{BC}\)[/tex] form a right angle at point [tex]\(B\)[/tex] and knowing the coordinates of points [tex]\(A\)[/tex] and [tex]\(B\)[/tex], we follow these steps:
1. Determine the direction vector of [tex]\(\overleftrightarrow{AB}\)[/tex]:
We have the coordinates: [tex]\(A = (-3, -1)\)[/tex] and [tex]\(B = (4, 4)\)[/tex].
The direction vector [tex]\(\overrightarrow{AB}\)[/tex] is calculated as:
[tex]\[
\overrightarrow{AB} = B - A = \begin{pmatrix} 4 - (-3) \\ 4 - (-1) \end{pmatrix} = \begin{pmatrix} 7 \\ 5 \end{pmatrix}
\][/tex]
2. Find the slope of [tex]\(\overleftrightarrow{AB}\)[/tex]:
The slope [tex]\(m_{AB}\)[/tex] is given by the change in [tex]\(y\)[/tex] over the change in [tex]\(x\)[/tex]:
[tex]\[
m_{AB} = \frac{\Delta y}{\Delta x} = \frac{5}{7}
\][/tex]
3. Determine the slope of [tex]\(\overleftrightarrow{BC}\)[/tex]:
Since [tex]\(\overleftrightarrow{BC}\)[/tex] is perpendicular to [tex]\(\overleftrightarrow{AB}\)[/tex], its slope [tex]\(m_{BC}\)[/tex] will be the negative reciprocal of [tex]\(m_{AB}\)[/tex]:
[tex]\[
m_{BC} = -\frac{1}{m_{AB}} = -\frac{7}{5}
\][/tex]
4. Use the point-slope form to find the equation of [tex]\(\overleftrightarrow{BC}\)[/tex]:
We will use point [tex]\(B = (4, 4)\)[/tex] and the slope [tex]\(m_{BC} = -\frac{7}{5}\)[/tex]. The point-slope form of the equation is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substituting [tex]\(x_1 = 4\)[/tex], [tex]\(y_1 = 4\)[/tex], and [tex]\(m = -\frac{7}{5}\)[/tex]:
[tex]\[
y - 4 = -\frac{7}{5}(x - 4)
\][/tex]
To eliminate the fraction, multiply through by 5:
[tex]\[
5(y - 4) = -7(x - 4)
\][/tex]
Distribute on both sides:
[tex]\[
5y - 20 = -7x + 28
\][/tex]
5. Rearrange to the standard form [tex]\(Ax + By = C\)[/tex]:
Move all terms involving variables to one side and constant terms to the other side:
[tex]\[
7x + 5y = 48
\][/tex]
Hence, the equation of the line [tex]\(\overleftrightarrow{BC}\)[/tex] is:
[tex]\[
7x + 5y = 48
\][/tex]
To match it with one of the given choices, it is important to observe the signs and structure. Multiplying both sides of [tex]\(7x + 5y = 48\)[/tex] by -1:
[tex]\[
-7x - 5y = -48
\][/tex]
And multiplying by -1 again reverses it to match the forms of one of the answer choices:
[tex]\[
7x - 5y = 48
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{D. \, 7 x - 5 y = 48}
\][/tex]
