Question

the point n lies on the segment mp. find the coordinates of n so that mn is $\frac{1}{8}$ of mp. p (30, 22) n (?,?) m (-2, -2) 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, $m = 1$ and $n=7$ (since $MN=\frac{1}{8}MP$, so the ratio of $MN$ to $NP$ is $1:7$), $x_1=-2,y_1 = - 2,x_2 = 30,y_2 = 22$.

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

$x=\frac{1\times30+7\times(-2)}{1 + 7}=\frac{30-14}{8}=\frac{16}{8}=2$.

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

$y=\frac{1\times22+7\times(-2)}{1 + 7}=\frac{22 - 14}{8}=\frac{8}{8}=1$.

Answer:

$(2,1)$