The answer to your question the equation of the line BC is C ![-7x - 5y = -48](/tpl/images/1402/0420/dafe8.png)
Since AB and BC form a right angle at point B, it means that AB is perpendicular to BC.
To find the equation of BC, we first need to find the gradient of BC.
Let m be the gradient of AB and m' be the gradient of BC, since AB is perpendicular to BC, mm' = -1. Thus m' = -1/m
So, we need to find the gradient of AB and thus find the gradient of BC.
To find the gradient of AB, we use the equation for the gradient of a line in slope-point form. Having point A = (x₁, y₁) = (-3, -1) and point B = (x₂, y₂) = (4, 4).
![m = \frac{y_{2} - y_{1} }{x_{2} - x_{1}}](/tpl/images/1402/0420/08b8e.png)
Substituting the values of the variables into the equation, we have
![m = \frac{4 - (-1) }{4 - (-3)}](/tpl/images/1402/0420/a58b8.png)
![m = \frac{4 + 1 }{4 + 3}](/tpl/images/1402/0420/c2a1b.png)
![m = \frac{5}{7}](/tpl/images/1402/0420/66e9c.png)
Since m' = -1/m
m' = -1 ÷ 5/7
m' = -7/5
Since we know the gradient of the line BC and the line BC passes through the point B = (4, 4) = (x₂, y₂) we find the equation of the line BC using the equation of a line in slope-point form.
![m' = \frac{y - y_{2} }{x - x_{2}}](/tpl/images/1402/0420/0d895.png)
![-\frac{7}{5} = \frac{y - 4 }{x - 4}](/tpl/images/1402/0420/5026d.png)
cross-multiplying, we have
![-7(x - 4) = 5(y - 4)](/tpl/images/1402/0420/f871e.png)
Expanding the brackets, we have
![-7x + 28 = 5y - 20](/tpl/images/1402/0420/b33a5.png)
Subtracting 28 from both sides, we have
![-7x = 5y - 48](/tpl/images/1402/0420/fcc2e.png)
Subtracting 5y from both sides, we have
![-7x - 5y = -48](/tpl/images/1402/0420/dafe8.png)
So, the equation of the line BC is ![-7x - 5y = -48](/tpl/images/1402/0420/dafe8.png)
The answer to your question the equation of the line BC is C ![-7x - 5y = -48](/tpl/images/1402/0420/dafe8.png)
Learn more about the equation of a line here: