Question

the coordinates of the endpoints of $overline{kl}$ are $k(-9,11)$ and $l(15, - 7)$. point $m$ is on $kl$ and divides it into two parts such that the ratio of the lengths is $1:2$. what are the two possible locations of $m? (-5,8) (-1,5) (3,2) (7,-1) (8,-2) (12,-5)$

Explanation:

Step1: Recall section - formula

If a point $M(x,y)$ divides the line - segment joining $K(x_1,y_1)$ and $L(x_2,y_2)$ in the ratio $m:n$, then the coordinates of $M$ are given by $x=\frac{mx_2+nx_1}{m + n}$ and $y=\frac{my_2+ny_1}{m + n}$. Here, $(x_1,y_1)=(-9,11)$ and $(x_2,y_2)=(15,-7)$, and we have two cases: $m = 1,n = 2$ and $m = 2,n = 1$.

Step2: Case 1: $m = 1,n = 2$

Calculate the $x$ - coordinate: $x=\frac{1\times15+2\times(-9)}{1 + 2}=\frac{15-18}{3}=\frac{-3}{3}=-1$.
Calculate the $y$ - coordinate: $y=\frac{1\times(-7)+2\times11}{1 + 2}=\frac{-7 + 22}{3}=\frac{15}{3}=5$. So, $M(-1,5)$.

Step3: Case 2: $m = 2,n = 1$

Calculate the $x$ - coordinate: $x=\frac{2\times15+1\times(-9)}{2 + 1}=\frac{30-9}{3}=\frac{21}{3}=7$.
Calculate the $y$ - coordinate: $y=\frac{2\times(-7)+1\times11}{2 + 1}=\frac{-14 + 11}{3}=\frac{-3}{3}=-1$. So, $M(7,-1)$.

Answer:

$(-1,5),(7,-1)$