Question

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

Answer 1

Step1: Calculation of slope of line through $\left(-2,3\right)$ and $\left(4,1\right)$:

The slope of a line passing through the points $(x_1,y_1)$ and $(x_2,y_2)$ is given by the formula $m=\frac{y_2-y_1}{x_2-x_1}$.

For the points $\left(-2,3\right)$ and $\left(4,1\right)$, $x_1=-2,y_1=3,x_2=4,y_2=1$

$$\begin{gathered} m=\frac{1-3}{4-(-2)} \\ \Rightarrow m=\frac{1-3}{4+2} \\ \Rightarrow m=\frac{-2}{6} \\ \therefore m=-\frac{1}{3} \end{gathered}$$

Step2: Calculation of the slope of line $3x=y+1$:

The slope-intercept form of the equation of a line is given by $y=mx+c$, where $m$ is the slope and $c$ is the $y$-intercept.

Simplify the equation to convert it into slope-intercept form.

$$\begin{gathered} 3x-y-1=0 \\ \Rightarrow 3x-1=y(addingytobothsides) \\ \therefore y=3x-1\left(rearranging\right) \end{gathered}$$

Comparing the equation (1) with equation $y=mx+c$ we get the slope of the line $3x-y-1=0$ as $m=3$.

The slope of line $3x-y-1=0$ i.e., $m=3$ is the negative reciprocal of the slope of line through $\left(-2,3\right)$ and $\left(4,1\right)$ i.e., $m=-\frac{1}{3}$.

Thus, the lines are perpendicular to each other.

Step3: Calculation of mid-point of line joining the points $\left(-2,3\right)$ and $\left(4,1\right)$.

The mid-point formula states that the co-ordinates of mid-point of the line joining the points $(x_1,y_1)$ and $(x_2,y_2)$ are given by the formula $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.

The perpendicular bisector of the line joining the points $\left(-2,3\right)$ and $\left(4,1\right)$. intersects it at the mid-point $C$ making an angle of $90°$.

Thus, by using the mid-point formula, the coordinates of point $C$ are:

$$\begin{gathered} x=\frac{-2+4}{2} \\ \Rightarrow x=\frac{2}{2} \\ \therefore x=1 \\ y=\frac{3+1}{2} \\ \Rightarrow y=\frac{4}{2} \\ \therefore y=2 \end{gathered}$$

Thus, the point of intersection of the line joining the points $\left(-2,3\right)$ and $\left(4,1\right)$ and its perpendicular bisector is $C\left(1,2\right)$.

Step4: Checking whether the line $3x=y+1$ bisect the join of $\left(-2,3\right)$ and $\left(4,1\right)$.

The line joining $\left(-2,3\right)$ and $\left(4,1\right)$ will bisect the line $3x=y+1$ if and only if the point $C\left(1,2\right)$ satisfies the equation of the line $3x=y+1$.

Substitute the point $C\left(1,2\right)$ in equation $3x=y+1$ and check whether the point satisfies the equation or not.

$$\begin{gathered} 3\left(1\right)=2+1 \\ \Rightarrow 3=3 \end{gathered}$$

Thus, the line $3x=y+1$ bisects the join of $\left(-2,3\right)$ and $\left(4,1\right)$.

Hence, the line through $\left(-2,3\right)$ and $\left(4,1\right)$ is perpendicular to the line $3x=y+1$ and yes, the line $3x=y+1$ bisects the join of $\left(-2,3\right)$ and $\left(4,1\right)$.