Question

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

Find the equation of the line through G and parallel to AC.

Answer 1

$\text{first let us find the coordinates of the centroid G(x, y)}$

$\text{By using the formula we can easily find coordinates of centroid;}x=\frac{x_1+x_2+x_3}{3};y=\frac{y_1+y_2+y_3}{3}$

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

$x=2;y=1$

$\text{Let us say the line passing through G and parallel to AC is:}y=mx+c$

$\text{where 'm' is the slope of the line a 'c' is constant}$

$\text{We can observe that the line is parallel to AC so slope of the lines are same,}$

$\text{so we can find 'm' by using the coordinates of A \& C}$

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{-2-3}{3-(-1)}=\frac{-5}{4}=-1.25$

$\text{our line should also pass through G, so let us put G into it to find the value of 'c'}$

$1=-1.25\left(2\right)+c$

$c=3.5$

$\text{So the final equation of line is:}y=-1.25x+3.5$