Question

the point n lies on the segment mp. find the coordinates of n so that mn is $\frac{2}{5}$ of mp. p (2,7) n (?,?) m (-13, -18) coordinates of n : ( , )

Explanation:

Step1: Use section - formula

If a point $N(x,y)$ divides the line - segment joining $M(x_1,y_1)$ and $P(x_2,y_2)$ in the ratio $m:n$, the coordinates of $N$ 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 $MN=\frac{2}{5}MP$, so the ratio of $MN$ to $NP$ is $2:3$), $x_1=-13$, $y_1=-18$, $x_2 = 2$, and $y_2 = 7$.

Step2: Calculate the x - coordinate of $N$

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

Step3: Calculate the y - coordinate of $N$

$y=\frac{2\times7+3\times(-18)}{2 + 3}=\frac{14-54}{5}=\frac{-40}{5}=-8$.

Answer:

$(-7,-8)$