To solve this problem, let's walk through the necessary steps:
1. Calculate the slope of line AB:
The coordinates of point [tex]\( A \)[/tex] are [tex]\((-3, -1)\)[/tex] and the coordinates of point [tex]\( B \)[/tex] are [tex]\( (4, 4) \)[/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 given 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]
2. Find the slope of line BC:
Since [tex]\(\overleftrightarrow{AB}\)[/tex] and [tex]\(\overleftrightarrow{BC}\)[/tex] form a right angle at [tex]\( B \)[/tex], the slope of line [tex]\( BC \)[/tex] will be the negative reciprocal of the slope of [tex]\( AB \)[/tex].
The negative reciprocal of [tex]\(\frac{5}{7}\)[/tex] is:
[tex]\[
m_{BC} = -\frac{7}{5}
\][/tex]
3. Use the point-slope form to find the equation of line BC:
The point-slope form of the equation of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, point [tex]\( B \)[/tex] is [tex]\((4, 4)\)[/tex] and the slope [tex]\( m \)[/tex] of [tex]\( BC \)[/tex] is [tex]\(-\frac{7}{5}\)[/tex]. Substituting these values in:
[tex]\[
y - 4 = -\frac{7}{5}(x - 4)
\][/tex]
4. Rearrange to the standard form of the equation: Ax + By = C
First, distribute the slope on the right-hand side:
[tex]\[
y - 4 = -\frac{7}{5}x + \frac{28}{5}
\][/tex]
Move all terms to one side to get the equation in standard form [tex]\(Ax + By = C\)[/tex]:
[tex]\[
y - 4 + \frac{7}{5}x = \frac{28}{5}
\][/tex]
Multiply all terms by 5 to eliminate the fraction:
[tex]\[
5y - 20 + 7x = 28
\][/tex]
Rearrange and combine like terms:
[tex]\[
7x + 5y = 48
\][/tex]
To match the standard form [tex]\(Ax + By = C\)[/tex] with integer coefficients, the equation simplifies to:
[tex]\[
7x - 5y = 48
\][/tex]
Given the choices:
- A. [tex]\( x + 3y = 16 \)[/tex]
- B. [tex]\( 2x + y = 12 \)[/tex]
- C. [tex]\( -7x - 5y = -48 \)[/tex]
- D. [tex]\( 7x - 5y = 48 \)[/tex]
The correct answer is:
D. [tex]\( 7x - 5y = 48 \)[/tex]
