To find the equation of the line [tex]\(\overleftrightarrow{BC}\)[/tex], we need to follow several steps.
1. Define points [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
- Point [tex]\(A\)[/tex] has coordinates [tex]\((-3, -1)\)[/tex],
- Point [tex]\(B\)[/tex] has coordinates [tex]\((4, 4)\)[/tex].
2. Calculate the direction vector of [tex]\(\overline{AB}\)[/tex]:
- The direction vector is found by subtracting the coordinates of [tex]\(A\)[/tex] from the coordinates of [tex]\(B\)[/tex]:
[tex]\[
AB = B - A = (4 - (-3), 4 - (-1)) = (4 + 3, 4 + 1) = (7, 5)
\][/tex]
3. Find a vector perpendicular to [tex]\(AB\)[/tex]:
- If [tex]\(AB = (x, y) = (7, 5)\)[/tex], a vector perpendicular to [tex]\(AB\)[/tex] can be [tex]\((-y, x)\)[/tex]:
[tex]\[
BC = (-5, 7)
\][/tex]
4. Determine the slope of the line [tex]\(BC\)[/tex]:
- The slope of the line [tex]\(BC\)[/tex] is given by the ratio of the vertical component to the horizontal component of the vector [tex]\(BC\)[/tex]:
[tex]\[
\text{slope of } BC = \frac{7}{-5} = -1.4
\][/tex]
5. Use the point-slope form of the equation of a line:
- The point-slope form is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\(m\)[/tex] is the slope.
6. Plug in point [tex]\(B = (4, 4)\)[/tex] and the slope:
- Using the slope [tex]\(m = -1.4\)[/tex] and point [tex]\(B(4, 4)\)[/tex]:
[tex]\[
y - 4 = -1.4(x - 4)
\][/tex]
7. Rearrange to get the slope-intercept form [tex]\(y = mx + c\)[/tex]:
[tex]\[
y - 4 = -1.4x + 5.6
\][/tex]
[tex]\[
y = -1.4x + 5.6 + 4
\][/tex]
[tex]\[
y = -1.4x + 9.6
\][/tex]
Thus, the equation of the line [tex]\(\overleftrightarrow{BC}\)[/tex] is:
[tex]\[
y = -1.4x + 9.6
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{y = -1.4x + 9.6}
\][/tex]
