Question

select the correct answer.
\\(\overleftrightarrow{ab}\\) and \\(\overleftrightarrow{bc}\\) form a right - angle at point b. if a = (-3, -1) and b = (4, 4), what is the equation of \\(\overleftrightarrow{bc}\\)?
a. x + 3y = 16
b. 2x + y = 12
c. -7x - 5y = -48
d. 7x - 5y = 48

Explanation:

Step1: Calculate slope of $\overrightarrow{AB}$

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Given $A(-3,-1)$ and $B(4,4)$, then $m_{AB}=\frac{4 - (-1)}{4-(-3)}=\frac{5}{7}$.

Step2: Find slope of $\overrightarrow{BC}$

Since $\overrightarrow{AB}$ and $\overrightarrow{BC}$ are perpendicular, the product of their slopes is - 1. Let the slope of $\overrightarrow{BC}$ be $m_{BC}$. Then $m_{AB}\times m_{BC}=-1$. So $m_{BC}=-\frac{7}{5}$.

Step3: Use point - slope form to find equation of $\overrightarrow{BC}$

The point - slope form of a line is $y - y_1=m(x - x_1)$. Using point $B(4,4)$ and $m =-\frac{7}{5}$, we have $y - 4=-\frac{7}{5}(x - 4)$.

Step4: Convert to standard form

$y-4=-\frac{7}{5}x+\frac{28}{5}$. Multiply through by 5 to get $5y-20=-7x + 28$. Rearranging gives $7x+5y = 48$.

Answer:

None of the given options are correct. The correct equation of $\overleftrightarrow{BC}$ should be $7x + 5y=48$. If there was a typo in option C and it was supposed to be $7x + 5y = 48$, then the answer would be C.