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: Find the slope of line 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: Determine the slope of line BC

Since $\overleftrightarrow{AB}$ and $\overleftrightarrow{BC}$ form a right - angle, the product of their slopes is $- 1$. Let the slope of $\overleftrightarrow{BC}$ be $m_{BC}$. Then $m_{AB}\times m_{BC}=-1$. Substituting $m_{AB}=\frac{5}{7}$, we get $m_{BC}=-\frac{7}{5}$.

Step3: Use the point - slope form to find the equation of line BC

The point - slope form of a line is $y - y_1=m(x - x_1)$. Using the point $B(4,4)$ and $m =-\frac{7}{5}$, we have $y - 4=-\frac{7}{5}(x - 4)$.
Expand: $y-4=-\frac{7}{5}x+\frac{28}{5}$.
Multiply through by 5 to clear the fraction: $5y-20=-7x + 28$.
Rearrange to the general form: $7x+5y = 48$.
We can also rewrite it in the form $-7x - 5y=-48$.

Answer:

C. $-7x - 5y=-48$