To find the equation of the line [tex]\(\overleftrightarrow{BC}\)[/tex] that forms a right angle with the line [tex]\(\overleftrightarrow{AB}\)[/tex] at point [tex]\(B\)[/tex], follow these steps:
1. Determine the slope of line [tex]\(\overleftrightarrow{AB}\)[/tex]:
The coordinates of points [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are given as [tex]\(A = (-3, -1)\)[/tex] and [tex]\(B = (4, 4)\)[/tex].
The formula to find 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]\[
m_{AB} = \frac{4 - (-1)}{4 - (-3)} = \frac{4 + 1}{4 + 3} = \frac{5}{7} \approx 0.7142857142857143
\][/tex]
2. Calculate the slope of line [tex]\(\overleftrightarrow{BC}\)[/tex]:
Since [tex]\(\overleftrightarrow{BC}\)[/tex] is perpendicular to [tex]\(\overleftrightarrow{AB}\)[/tex] at point [tex]\(B\)[/tex], the slope of [tex]\(\overleftrightarrow{BC}\)[/tex] will be the negative reciprocal of the slope of [tex]\(\overleftrightarrow{AB}\)[/tex]. If [tex]\(m_{AB}\)[/tex] is the slope of [tex]\(\overleftrightarrow{AB}\)[/tex], then the slope [tex]\(m_{BC}\)[/tex] of [tex]\(\overleftrightarrow{BC}\)[/tex] is:
[tex]\[
m_{BC} = -\frac{1}{m_{AB}} = -\frac{1}{0.7142857142857143} \approx -1.4
\][/tex]
3. Find the y-intercept of the line [tex]\(\overleftrightarrow{BC}\)[/tex]:
Using the point-slope form of a line equation [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\((x_1, y_1)\)[/tex] is the point [tex]\(B (4, 4)\)[/tex] and [tex]\(m\)[/tex] is the slope [tex]\(m_{BC}\)[/tex]:
[tex]\[
y - 4 = -1.4(x - 4)
\][/tex]
Simplify and solve for [tex]\(y\)[/tex] to get the equation in 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]
4. State the equation of line [tex]\(\overleftrightarrow{BC}\)[/tex]:
The equation of the line [tex]\(\overleftrightarrow{BC}\)[/tex] is:
[tex]\[
y = -1.4x + 9.6
\][/tex]
So, the correct equation of line [tex]\(\overleftrightarrow{BC}\)[/tex] is:
[tex]\[
y = -1.4x + 9.6
\][/tex]
