To determine the equation of the line [tex]\( \overleftrightarrow{BC} \)[/tex], given that [tex]\( \overleftrightarrow{AB} \)[/tex] and [tex]\( \overleftrightarrow{BC} \)[/tex] form a right angle at point [tex]\( B \)[/tex], and the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are [tex]\( A = (-3, -1) \)[/tex] and [tex]\( B = (4, 4) \)[/tex], follow these steps:
1. Calculate the slope of [tex]\( \overleftrightarrow{AB} \)[/tex]:
The slope [tex]\( m \)[/tex] of a line through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting [tex]\(A\)[/tex] and [tex]\(B\)[/tex] coordinates:
[tex]\[
m_{AB} = \frac{4 - (-1)}{4 - (-3)} = \frac{4 + 1}{4 + 3} = \frac{5}{7}
\][/tex]
2. Determine the slope of [tex]\( \overleftrightarrow{BC} \)[/tex]:
Because [tex]\( \overleftrightarrow{AB} \)[/tex] and [tex]\( \overleftrightarrow{BC} \)[/tex] are perpendicular, the slope [tex]\( m \)[/tex] of [tex]\( \overleftrightarrow{BC} \)[/tex] is the negative reciprocal of the slope of [tex]\( \overleftrightarrow{AB} \)[/tex]:
[tex]\[
m_{BC} = -\frac{1}{m_{AB}} = -\frac{1}{\frac{5}{7}} = -\frac{7}{5}
\][/tex]
3. Find the equation of the line [tex]\( \overleftrightarrow{BC} \)[/tex]:
The equation of a line in point-slope form 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]
4. Convert to slope-intercept form:
Solving for [tex]\( y \)[/tex]:
[tex]\[
y - 4 = -\frac{7}{5} x + \frac{28}{5}
\][/tex]
[tex]\[
y = -\frac{7}{5}x + \frac{28}{5} + 4
\][/tex]
[tex]\[
y = -\frac{7}{5}x + \frac{28}{5} + \frac{20}{5}
\][/tex]
[tex]\[
y = -\frac{7}{5}x + \frac{48}{5}
\][/tex]
5. Convert to general form [tex]\( ax + by = c \)[/tex]:
Multiply through by 5 to clear the fraction:
[tex]\[
5y = -7x + 48
\][/tex]
Rearrange to general form:
[tex]\[
7x + 5y = 48
\][/tex]
Compare to the given options:
- Option A: [tex]\( x + 3y = 16 \)[/tex]
- Option B: [tex]\( 2x + y = 12 \)[/tex]
- Option C: [tex]\( -7x - 5y = -48 \)[/tex]
- Option D: [tex]\( 7x - 5y = 48 \)[/tex]
None of the options match [tex]\( 7x + 5y = 48 \)[/tex].
Therefore, the equation of [tex]\( \overleftrightarrow{BC} \)[/tex], given the points and right angle at [tex]\( B \)[/tex], is confirmed by running through the correct procedures and calculations as:
[tex]\[
\boxed{\text{None}}
\][/tex]
