Select the correct answer.

[tex]\overleftrightarrow{AB}[/tex] and [tex]\overleftrightarrow{BC}[/tex] form a right angle at point [tex]B[/tex]. If [tex]A=(-3, -1)[/tex] and [tex]B=(4, 4)[/tex], what is the equation of [tex]\overleftrightarrow{BC}[/tex]?

A. [tex]x + 3y = 16[/tex]

B. [tex]2x + y = 12[/tex]

C. [tex]-7x - 5y = -48[/tex]

D. [tex]7x - 5y = 48[/tex]



Answer :

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]