Question

the point b lies on the segment $overline{ac}$. find the coordinates of b so that ab is $\frac{2}{5}$ of ac. a (-13, 18) b (?,?) c (2, -7)

Explanation:

Step1: Use the section - formula

If a point $B(x,y)$ divides the line - segment joining $A(x_1,y_1)$ and $C(x_2,y_2)$ in the ratio $m:n$, the coordinates of $B$ are given by $x=\frac{mx_2+nx_1}{m + n}$ and $y=\frac{my_2+ny_1}{m + n}$. Here, $m = 2$, $n=3$ (since $AB=\frac{2}{5}AC$, so the ratio of $AB$ to $BC$ is $2:3$), $x_1=-13$, $y_1 = 18$, $x_2=2$, and $y_2=-7$.

Step2: Calculate the $x$ - coordinate of $B$

$x=\frac{2\times2+3\times(-13)}{2 + 3}=\frac{4-39}{5}=\frac{-35}{5}=-7$.

Step3: Calculate the $y$ - coordinate of $B$

$y=\frac{2\times(-7)+3\times18}{2 + 3}=\frac{-14 + 54}{5}=\frac{40}{5}=8$.

Answer:

$(-7,8)$