To determine the equation of the line through point [tex]\( B \)[/tex] (4, 4) that is perpendicular to the line passing through points [tex]\( A \)[/tex] (-3, -1) and [tex]\( B \)[/tex] (4, 4), follow these steps:
1. Calculate the slope of line [tex]\( AB \)[/tex]:
The formula for the slope 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]
For 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 the line perpendicular to [tex]\( AB \)[/tex]:
The slope of a line perpendicular to another is the negative reciprocal of the original slope. Thus, the perpendicular slope [tex]\( m_{BC} \)[/tex] is:
[tex]\[
m_{BC} = -\frac{1}{m_{AB}} = -\frac{1}{\frac{5}{7}} = -\frac{7}{5}
\][/tex]
3. Write the equation of line [tex]\( BC \)[/tex] in point-slope form using point [tex]\( B \)[/tex]:
The point-slope form of a line is:
[tex]\[
y - y_1 = m (x - x_1)
\][/tex]
Substituting [tex]\( m_{BC} = -\frac{7}{5} \)[/tex] and point [tex]\( B \)[/tex] (4, 4):
[tex]\[
y - 4 = -\frac{7}{5} (x - 4)
\][/tex]
4. Convert to slope-intercept form [tex]\( y = mx + c \)[/tex]:
Expanding the equation:
[tex]\[
y - 4 = -\frac{7}{5}x + \frac{28}{5}
\][/tex]
Adding 4 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = -\frac{7}{5}x + \frac{28}{5} + 4
\][/tex]
Converting 4 to a fraction with a common denominator:
[tex]\[
4 = \frac{20}{5}
\][/tex]
So:
[tex]\[
y = -\frac{7}{5}x + \frac{28}{5} + \frac{20}{5} = -\frac{7}{5}x + \frac{48}{5}
\][/tex]
5. Convert to the general form [tex]\( Ax + By = C \)[/tex]:
Multiply every term by 5 to eliminate the fractions:
[tex]\[
5y = -7x + 48
\][/tex]
Rearranging terms:
[tex]\[
7x + 5y = 48
\][/tex]
Thus, the equation of the line [tex]\( \overleftrightarrow{B C} \)[/tex] is:
[tex]\[
7x - 5y = 48
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{7x - 5y = 48}
\][/tex]
