Let's solve the problem step by step.
1. Identify the coordinates of points A and B:
- [tex]\( A = (-3, -1) \)[/tex]
- [tex]\( B = (4, 4) \)[/tex]
2. Calculate the slope of line AB:
- The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
- Using the coordinates of A and B:
[tex]\[
m_{AB} = \frac{4 - (-1)}{4 - (-3)} = \frac{4 + 1}{4 + 3} = \frac{5}{7}
\][/tex]
3. Determine the slope of line BC:
- Since [tex]\(\overleftrightarrow{A B}\)[/tex] and [tex]\(\overleftrightarrow{B C}\)[/tex] form a right angle at point [tex]\( B \)[/tex], the slopes [tex]\( m_{AB} \)[/tex] and [tex]\( m_{BC} \)[/tex] are negative reciprocals of each other:
[tex]\[
m_{BC} = -\frac{1}{m_{AB}} = -\frac{1}{\frac{5}{7}} = -\frac{7}{5}
\][/tex]
4. Equation of line BC using point-slope form:
- The point-slope form of the equation of a line is [tex]\( y - y_1 = m(x - x_1) \)[/tex].
- Using point [tex]\( B(4, 4) \)[/tex] and [tex]\( m_{BC} = -\frac{7}{5} \)[/tex]:
[tex]\[
y - 4 = -\frac{7}{5}(x - 4)
\][/tex]
5. Convert the equation to the general form [tex]\( Ax + By = C \)[/tex]:
- Distribute the slope on the right side:
[tex]\[
y - 4 = -\frac{7}{5}x + \frac{28}{5}
\][/tex]
- Multiply through by 5 to clear the fraction:
[tex]\[
5y - 20 = -7x + 28
\][/tex]
- Rearrange the terms to get:
[tex]\[
7x + 5y = 48
\][/tex]
6. Compare with the given options:
- A. [tex]\( x + 3y = 16 \)[/tex]
- B. [tex]\( 2x + y = 12 \)[/tex]
- C. [tex]\( -7x - 5y = -48 \)[/tex]
- D. [tex]\( 7x - 5y = 48 \)[/tex]
Based on our derived equation [tex]\( 7x + 5y = 48 \)[/tex], the correct answer is not directly among the given choices. However, if we can verify any of the options might be misread or altered in the problem or options, that can give hints.
Since we are expected to choose based only on derived equation step consistency:
Thus, reviewing our derived results [tex]\(\boxed{None}\)[/tex]
