Question

find the coordinates of the point $\frac{1}{3}$ of the way from $a$ to $b$.
$a(-3,-2)$
$b(12,4)$
coordinates = ( )

Explanation:

Step1: Use the section - formula for x - coordinate

The formula for the x - coordinate of a point that divides the line segment joining $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m:n$ is $x=\frac{mx_2+nx_1}{m + n}$. Here, $m = 1$, $n=2$, $x_1=-3$, $x_2 = 12$. So, $x=\frac{1\times12+2\times(-3)}{1 + 2}$.
$x=\frac{12-6}{3}=\frac{6}{3}=2$.

Step2: Use the section - formula for y - coordinate

The formula for the y - coordinate of a point that divides the line segment joining $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m:n$ is $y=\frac{my_2+ny_1}{m + n}$. Here, $m = 1$, $n = 2$, $y_1=-2$, $y_2=4$. So, $y=\frac{1\times4+2\times(-2)}{1 + 2}$.
$y=\frac{4 - 4}{3}=0$.

Answer:

$(2,0)$