To find the equation of the line [tex]\( \overleftrightarrow{BC} \)[/tex] that forms a right angle with [tex]\( \overleftrightarrow{AB} \)[/tex] at point [tex]\( B \)[/tex], and given the points [tex]\( A = (-3, -1) \)[/tex] and [tex]\( B = (4, 4) \)[/tex], follow these steps:
1. Calculate the slope of line [tex]\( AB \)[/tex]:
To find the slope of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], use the formula:
[tex]\[
\text{slope}_{AB} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting in the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
\text{slope}_{AB} = \frac{4 - (-1)}{4 - (-3)} = \frac{4 + 1}{4 + 3} = \frac{5}{7}
\][/tex]
2. Determine the slope of line [tex]\( BC \)[/tex]:
Since lines [tex]\( \overleftrightarrow{AB} \)[/tex] and [tex]\( \overleftrightarrow{BC} \)[/tex] form a right angle, the slope of [tex]\( BC \)[/tex], let's call it [tex]\( m_{BC} \)[/tex], is the negative reciprocal of the slope of [tex]\( AB \)[/tex]. Thus:
[tex]\[
m_{BC} = -\frac{1}{\text{slope}_{AB}} = -\frac{1}{\frac{5}{7}} = -\frac{7}{5}
\][/tex]
3. Find the equation of line [tex]\( BC \)[/tex] using the point-slope form of the equation of a line:
The point-slope form is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\((x_1, y_1)\)[/tex] is point [tex]\( B \)[/tex] [tex]\((4, 4)\)[/tex] and [tex]\( m \)[/tex] is the slope of the line [tex]\( BC \)[/tex] which is [tex]\(-\frac{7}{5}\)[/tex]:
[tex]\[
y - 4 = -\frac{7}{5}(x - 4)
\][/tex]
To clear the fraction, multiply every term by 5:
[tex]\[
5(y - 4) = -7(x - 4)
\][/tex]
Simplify:
[tex]\[
5y - 20 = -7x + 28
\][/tex]
Rearrange to a more standard linear equation form [tex]\( ax + by = c \)[/tex]:
[tex]\[
7x + 5y = 48
\][/tex]
So, the equation of the line [tex]\( \overleftrightarrow{BC} \)[/tex] is:
[tex]\[
7x + 5y = 48
\][/tex]
Finally, match this result with the provided answer choices.
The correct answer is:
[tex]\[
\boxed{7 x + 5 y = 48}
\][/tex]
