To find the equation of the line [tex]\(\overleftrightarrow{BC}\)[/tex] which is perpendicular to the line [tex]\(\overleftrightarrow{AB}\)[/tex] and passes through the point [tex]\(B = (4, 4)\)[/tex], follow these steps:
1. Find the Slope of Line [tex]\(\overleftrightarrow{AB}\)[/tex]:
Line [tex]\(\overleftrightarrow{AB}\)[/tex] passes through the points [tex]\(A = (-3, -1)\)[/tex] and [tex]\(B = (4, 4)\)[/tex].
The slope [tex]\(m_{AB}\)[/tex] of [tex]\(\overleftrightarrow{AB}\)[/tex] is given by:
[tex]\[
m_{AB} = \frac{y_B - y_A}{x_B - x_A} = \frac{4 - (-1)}{4 - (-3)} = \frac{4 + 1}{4 + 3} = \frac{5}{7}
\][/tex]
2. Find the Slope of Line [tex]\(\overleftrightarrow{BC}\)[/tex]:
Since [tex]\(\overleftrightarrow{BC}\)[/tex] is perpendicular to [tex]\(\overleftrightarrow{AB}\)[/tex], its slope [tex]\(m_{BC}\)[/tex] is the negative reciprocal of [tex]\(m_{AB}\)[/tex]:
[tex]\[
m_{BC} = -\frac{1}{m_{AB}} = -\frac{1}{\frac{5}{7}} = -\frac{7}{5}
\][/tex]
3. Write the Equation of Line [tex]\(\overleftrightarrow{BC}\)[/tex]:
We use the point-slope form of the equation of a line, [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\((x_1, y_1) = (4, 4)\)[/tex] and [tex]\(m = m_{BC} = -\frac{7}{5}\)[/tex].
Substituting the values, we get:
[tex]\[
y - 4 = -\frac{7}{5}(x - 4)
\][/tex]
4. Convert the Equation to Standard Form:
Simplify and convert the above equation to the standard form [tex]\(Ax + By = C\)[/tex].
First, distribute the slope:
[tex]\[
y - 4 = -\frac{7}{5}x + \frac{7}{5} \cdot 4
\][/tex]
[tex]\[
y - 4 = -\frac{7}{5}x + \frac{28}{5}
\][/tex]
Multiply through by 5 to clear the fraction:
[tex]\[
5(y - 4) = 5\left(-\frac{7}{5}x + \frac{28}{5}\right)
\][/tex]
[tex]\[
5y - 20 = -7x + 28
\][/tex]
Rearrange to get the standard form:
[tex]\[
7x - 5y = 48
\][/tex]
Thus, the equation of the line [tex]\(\overleftrightarrow{BC}\)[/tex] is:
[tex]\[
7x - 5y = 48
\][/tex]
So, the correct answer is:
D. [tex]\(7 x - 5 y = 48\)[/tex]
