Question

Is the line through $(-2,3)$ and $(4,1)$ perpendicular to the line $3x=y+1$ ?
Does the line $3x=y+1$ bisect the join of $(-2,3)$ and $(4,1)$.

Answer 1

Slope of the line passing through the points $(-2,3)$ and $(4,1)$ is given by

$m_1=\frac{y_2-y_1}{x_2-x_1}$

$=\frac{(1-3)}{(4+2)}$

$=\frac{-2}{6}$

$=\frac{-1}{3}$

And, the slope of the line : $3x=y+1$

$y=3x-1$

Slope $\left(m_2\right)=3$

Now,

$m_1\times m_2=\frac{-1}{3}\times 3=-1$

Thus, the lines are perpendicular to each other as the product of their slopes is $-1$.

Now,

Co-ordinates of the mid - point of the line joining the points $(-2,3)$ and $(4,1)$ is

$(\frac{[-2+4]}{2},\frac{[3+1]}{2})=(1,2)$

Now, if the line $3x=y+1$ passes through the mid - point then it will satisfy the equation

$3\left(1\right)=\left(2\right)+1$

$3=3$

Hence, the line $3x=y+1$ bisects the line joining the points $(-2,3)$ and $(4,1)$.