To determine the equation of the line \(\overleftrightarrow{BC}\) which forms a right angle with the line \(\overleftrightarrow{AB}\) at point \(B\), we can follow these steps:
1. Calculate the slope of \(\overleftrightarrow{AB}\):
Given points \(A(-3, -1)\) and \(B(4, 4)\), the slope \(m_{AB}\) is calculated as:
[tex]\[
m_{AB} = \frac{4 - (-1)}{4 - (-3)} = \frac{4 + 1}{4 + 3} = \frac{5}{7}
\][/tex]
2. Find the slope of \(\overleftrightarrow{BC}\):
Since \(\overleftrightarrow{BC}\) forms a right angle with \(\overleftrightarrow{AB}\), the slope \(m_{BC}\) is the negative reciprocal of \(m_{AB}\):
[tex]\[
m_{BC} = -\frac{1}{m_{AB}} = -\frac{1}{\frac{5}{7}} = -\frac{7}{5}
\][/tex]
3. Write the equation of the line \(\overleftrightarrow{BC}\) in slope-intercept form:
Using point \(B(4, 4)\) and the slope \(m_{BC} = -\frac{7}{5}\), we use the point-slope form equation \(y - y_1 = m(x - x_1)\):
[tex]\[
y - 4 = -\frac{7}{5}(x - 4)
\][/tex]
Distributing the slope and simplifying:
[tex]\[
y - 4 = -\frac{7}{5}x + \frac{28}{5}
\][/tex]
[tex]\[
y = -\frac{7}{5}x + \frac{28}{5} + 4
\][/tex]
Converting 4 into a fraction with denominator 5:
[tex]\[
y = -\frac{7}{5}x + \frac{28}{5} + \frac{20}{5}
\][/tex]
[tex]\[
y = -\frac{7}{5}x + \frac{48}{5}
\][/tex]
4. Convert to standard form \(Ax + By = C\):
To eliminate the fractions, multiply every term by 5:
[tex]\[
5y = -7x + 48
\][/tex]
[tex]\[
7x + 5y = 48
\][/tex]
Now compare the converted equation \(7x + 5y = 48\) with the given options:
A. \( x + 3y = 16 \)
B. \( 2x + y = 12 \)
C. \(-7x - 5y = -48 \)
D. \(7x - 5y = 48 \)
The correct equation that matches is:
[tex]\[
\boxed{7x - 5y = 48}
\][/tex]
Option D is the correct answer.
