To find the equation of the line [tex]\(\overleftrightarrow{BC}\)[/tex] that forms a right angle with the line [tex]\(\overleftrightarrow{AB}\)[/tex] at point [tex]\(B\)[/tex], we need to follow these steps:
1. Find the slope of [tex]\(\overleftrightarrow{AB}\)[/tex]:
Given the points [tex]\(A = (-3, -1)\)[/tex] and [tex]\(B = (4, 4)\)[/tex], we use the formula for the slope [tex]\(m\)[/tex]:
[tex]\[
m_{AB} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the given points [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[
m_{AB} = \frac{4 - (-1)}{4 - (-3)} = \frac{4 + 1}{4 + 3} = \frac{5}{7}
\][/tex]
2. 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]\[
m_{BC} = -\frac{1}{m_{AB}} = -\frac{1}{\frac{5}{7}} = -\frac{7}{5}
\][/tex]
3. Use point-slope form of the line equation:
We know the slope of [tex]\(\overleftrightarrow{BC}\)[/tex] and it passes through [tex]\(B = (4, 4)\)[/tex]. The point-slope form of the line equation is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substituting [tex]\(m = -\frac{7}{5}\)[/tex] and the coordinates of point [tex]\(B\)[/tex]:
[tex]\[
y - 4 = -\frac{7}{5}(x - 4)
\][/tex]
4. Convert to general form [tex]\(Ax + By = C\)[/tex]:
Simplify and rearrange the equation to standard form:
[tex]\[
y - 4 = -\frac{7}{5}(x - 4)
\][/tex]
Multiply both sides by 5 to eliminate the fraction:
[tex]\[
5(y - 4) = -7(x - 4)
\][/tex]
Distribute and simplify:
[tex]\[
5y - 20 = -7x + 28
\][/tex]
Bring all terms to one side to achieve the general form [tex]\(Ax + By = C\)[/tex]:
[tex]\[
7x - 5y = 48
\][/tex]
Thus, the equation of the line [tex]\(\overleftrightarrow{BC}\)[/tex] is:
[tex]\[
7x - 5y = 48
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{D \, \text{ 7x - 5y = 48}}
\][/tex]
