Question

the point n lies on the segment mp. find the coordinates of n so that mn is $\frac{1}{4}$ of mp. m (-28, 22) n (?,?) p (4, -2) coordinates of n : (, )

Explanation:

Step1: Use section - formula

The section - formula for a point $N(x,y)$ that divides the line - segment joining $M(x_1,y_1)$ and $P(x_2,y_2)$ in the ratio $m:n$ is given by $x=\frac{mx_2+nx_1}{m + n}$ and $y=\frac{my_2+ny_1}{m + n}$. Here, $m = 1$ and $n=3$ (since $MN=\frac{1}{4}MP$, so the ratio of $MN$ to $NP$ is $1:3$), $x_1=-28$, $y_1 = 22$, $x_2 = 4$, and $y_2=-2$.

Step2: Calculate the x - coordinate of N

$x=\frac{1\times4+3\times(-28)}{1 + 3}=\frac{4-84}{4}=\frac{-80}{4}=-20$.

Step3: Calculate the y - coordinate of N

$y=\frac{1\times(-2)+3\times22}{1 + 3}=\frac{-2 + 66}{4}=\frac{64}{4}=16$.

Answer:

$(-20,16)$