Question

the point n lies on the segment $overline{mp}$. find the coordinates of n so that mn is $\frac{7}{9}$ of mp. m (-3,4) n (?,?) p (24,-12) coordinates of n: ( )

Explanation:

Step1: Use the 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 $m:n$, then $x=\frac{mx_2+nx_1}{m + n}$ and $y=\frac{my_2+ny_1}{m + n}$. Here, $m = 7$, $n=9 - 7=2$, $x_1=-3$, $y_1 = 4$, $x_2 = 24$, and $y_2=-12$.

Step2: Calculate the x - coordinate of N

$x=\frac{7\times24+2\times(-3)}{7 + 2}=\frac{168-6}{9}=\frac{162}{9}=18$.

Step3: Calculate the y - coordinate of N

$y=\frac{7\times(-12)+2\times4}{7 + 2}=\frac{-84 + 8}{9}=\frac{-76}{9}=-\frac{76}{9}$.

Answer:

$(18,-\frac{76}{9})$