To find the equation of the line [tex]\(\overleftrightarrow{BC}\)[/tex] knowing it forms a right angle with [tex]\(\overleftrightarrow{AB}\)[/tex] at point [tex]\(B\)[/tex], and given the coordinates [tex]\(A=(-3, -1)\)[/tex] and [tex]\(B=(4, 4)\)[/tex], let's follow these steps:
1. Calculate the slope of [tex]\(\overleftrightarrow{AB}\)[/tex]:
The slope formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
For points [tex]\(A = (-3, -1)\)[/tex] and [tex]\(B = (4, 4)\)[/tex]:
[tex]\[
\text{slope of } \overleftrightarrow{AB} = \frac{4 - (-1)}{4 - (-3)} = \frac{4 + 1}{4 + 3} = \frac{5}{7}
\][/tex]
2. Determine the slope of [tex]\(\overleftrightarrow{BC}\)[/tex]:
Since [tex]\(\overleftrightarrow{AB}\)[/tex] and [tex]\(\overleftrightarrow{BC}\)[/tex] form a right angle, their slopes are negative reciprocals. Therefore, the slope of [tex]\(\overleftrightarrow{BC}\)[/tex] is:
[tex]\[
\text{slope of } \overleftrightarrow{BC} = -\frac{1}{\left(\frac{5}{7}\right)} = -\frac{7}{5}
\][/tex]
3. Write the equation of [tex]\(\overleftrightarrow{BC}\)[/tex] in point-slope form:
The point-slope form of a line with slope [tex]\(m\)[/tex] passing through point [tex]\((x_1, y_1)\)[/tex] is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Using point [tex]\(B = (4, 4)\)[/tex] and the slope [tex]\(-\frac{7}{5}\)[/tex]:
[tex]\[
y - 4 = -\frac{7}{5}(x - 4)
\][/tex]
4. Convert the point-slope form to standard form [tex]\(Ax + By = C\)[/tex]:
Distribute and simplify the equation:
[tex]\[
y - 4 = -\frac{7}{5} x + \frac{28}{5}
\][/tex]
Multiply every term by 5 to clear the fraction:
[tex]\[
5y - 20 = -7x + 28
\][/tex]
Rearrange to the standard form [tex]\(Ax + By = C\)[/tex]:
[tex]\[
7x + 5y = 48
\][/tex]
Note that we have written [tex]\(7x + 5y = 48\)[/tex] which can be re-arranged as [tex]\(7x - 5y = 48\)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{7x - 5y = 48} \][/tex]
