Question

$A(-1,3),B(4,2)andC(3,-2)$ are the vertices of a triangle .

(i) Find the coordinates of the centroid $G$ of the triangle.

(ii) Find the equation of the line through $G$ and parallel to $AC$.

Answer 1

Step1- Finding the coordinates of the centroid $G$ of the triangle:

Given points are,$A(-1,3),B(4,2)andC(3,-2)$

The co-ordinate of centroid is given by $G(x,y)=\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)$

$\Rightarrow G(x,y)=\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)$

$$\begin{gathered} \Rightarrow G(x,y)=\left(\frac{-1+4+3}{3},\frac{3+2-2}{3}\right) \\ \Rightarrow G(x,y)=\left(\frac{6}{3},\frac{3}{3}\right) \\ \Rightarrow G(x,y)=\left(2,1\right) \end{gathered}$$

Hence, the coordinates of the centroid $G$ of the triangle is $\left(2,1\right)$.

Step 2- Find the equation of the line through $G$ and parallel to $AC$.:

Hence, the coordinates of the centroid $G$ of the triangle is $\left(2,1\right)$.

Now , as the required line is parallel to $AC$ , then its slope will be

$$\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ \Rightarrow m=\frac{-2-3}{3+1} \\ \Rightarrow m=\frac{-5}{4} \end{gathered}$$

As ,when two lines are parallel to each other then they both have same slope ,

Since, the required line is parallel to $AC$ , then its slope will be $\frac{-5}{4}$

So, the equation of line passing through $G$ and having slope as $\frac{-5}{4}$ is ,

$$\begin{gathered} y-1=\frac{-5}{4}(x-2) \\ 4y-4=-5x+10 \\ 5x+4y-4-10=0 \\ 5x+4y=14 \end{gathered}$$

Hence , the equation of line passing through $G$ $\left(2,1\right)$ and parallel to $AC$ is $5x+4y=14$.