Certainly! Let's solve the problem step-by-step to find the equation of line [tex]\(\overleftrightarrow{B C}\)[/tex].
1. Identify the given points:
- Point [tex]\(A\)[/tex] is [tex]\((-3, -1)\)[/tex].
- Point [tex]\(B\)[/tex] is [tex]\((4, 4)\)[/tex].
2. Find the slope of line [tex]\(\overleftrightarrow{A B}\)[/tex]:
- 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].
- Substituting the coordinates of 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]
3. Determine the slope of line [tex]\(\overleftrightarrow{B C}\)[/tex]:
- Lines [tex]\(\overleftrightarrow{A B}\)[/tex] and [tex]\(\overleftrightarrow{B C}\)[/tex] form a right angle at point [tex]\(B\)[/tex].
- Therefore, the slopes of these lines are negative reciprocals of each other.
- Hence, the slope [tex]\(m_{BC}\)[/tex] of line [tex]\(\overleftrightarrow{B C}\)[/tex] is:
[tex]\[
m_{BC} = -\frac{1}{m_{AB}} = -\frac{1}{\frac{5}{7}} = -\frac{7}{5}
\][/tex]
4. Form the equation of the line [tex]\(\overleftrightarrow{B C}\)[/tex] using point-slope form:
- The point-slope form of a line's equation is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
- Substituting the slope [tex]\(m_{BC} = -\frac{7}{5}\)[/tex] and the coordinates of point [tex]\(B\)[/tex] [tex]\((4, 4)\)[/tex]:
[tex]\[
y - 4 = -\frac{7}{5}(x - 4)
\][/tex]
5. Simplify the equation to general form:
- Expand the equation:
[tex]\[
y - 4 = -\frac{7}{5}x + \frac{28}{5}
\][/tex]
- Multiply every term by 5 to eliminate the fraction:
[tex]\[
5(y - 4) = 5 \left( -\frac{7}{5}x + \frac{28}{5} \right)
\][/tex]
[tex]\[
5y - 20 = -7x + 28
\][/tex]
[tex]\[
7x + 5y = 48
\][/tex]
Therefore, the equation of line [tex]\(\overleftrightarrow{B C}\)[/tex] is:
[tex]\[
7x - 5y = 48
\][/tex]
The correct answer is:
[tex]\[
\boxed{7 x - 5 y = 48}
\][/tex]
