To determine the equation of the line [tex]\(\overleftrightarrow{BC}\)[/tex] that forms a right angle with [tex]\(\overleftrightarrow{A B}\)[/tex] at point [tex]\(B\)[/tex], follow these steps:
1. Calculate the slope of line [tex]\(\overleftrightarrow{A B}\)[/tex]:
Line [tex]\(\overleftrightarrow{A B}\)[/tex] is defined by the points [tex]\(A(-3, -1)\)[/tex] and [tex]\(B(4, 4)\)[/tex]:
[tex]\[
\text{slope}_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-1)}{4 - (-3)} = \frac{5}{7} \approx 0.7143
\][/tex]
2. Find the slope of line [tex]\(\overleftrightarrow{BC}\)[/tex] that is perpendicular to [tex]\(\overleftrightarrow{A B}\)[/tex]:
For two lines to be perpendicular, the product of their slopes must be [tex]\(-1\)[/tex]. Hence, if [tex]\(m_{AB}\)[/tex] is the slope of [tex]\(\overleftrightarrow{A B}\)[/tex] ([tex]\(0.7143\)[/tex]), then the slope [tex]\(m_{BC}\)[/tex] of [tex]\(\overleftrightarrow{BC}\)[/tex] is the negative reciprocal of [tex]\(m_{AB}\)[/tex]:
[tex]\[
m_{BC} = -\frac{1}{m_{AB}} = -\frac{1}{0.7143} \approx -1.4
\][/tex]
3. Use the point-slope form of the line equation to find the equation of [tex]\(\overleftrightarrow{BC}\)[/tex]:
The point-slope form of a line's equation is given by [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. Since [tex]\(\overleftrightarrow{BC}\)[/tex] passes through point [tex]\(B(4, 4)\)[/tex]:
[tex]\[
y - 4 = -1.4(x - 4)
\][/tex]
4. Simplify the equation to the standard form [tex]\(Ax + By = C\)[/tex]:
Distribute [tex]\( -1.4 \)[/tex]:
[tex]\[
y - 4 = -1.4x + 5.6
\][/tex]
Add 4 to both sides to isolate [tex]\(y\)[/tex]:
[tex]\[
y = -1.4x + 5.6 + 4
\][/tex]
[tex]\[
y = -1.4x + 9.6
\][/tex]
Rearrange the equation to get it into [tex]\(Ax + By = C\)[/tex] form by moving all terms to one side:
[tex]\[
1.4x + y = 9.6
\][/tex]
5. Identify the correct answer from the provided options:
Comparing this equation [tex]\(1.4x + y = 9.6\)[/tex] to the given choices, the correct equation corresponds to none directly. However, examining the choices more, option A represents the closest in transformed form equivalents (properly scaled up or down).
Hence, the equation for [tex]\(\overleftrightarrow{BC}\)[/tex] forming a right angle with [tex]\(\overleftrightarrow{A B}\)[/tex] is:
[tex]\[
1.4x + y = 9.6
\][/tex]
Therefore:
The answer is not one provided in direct matches but from constructed equivalents, the closest proper form.
But strictly here none mentioned matches, yet standard simplifications outlies nearest.
