Question

the coordinates of the endpoints of $overline{lm}$ are $l(-7, -7)$ and $m(5, 9)$. point $n$ is on $overline{lm}$ and divides it such that $ln:mn$ is $1:3$. what are the coordinates of $n$? write your answers as integers or decimals.

Explanation:

Step1: Use the section - formula for x - coordinate

The section formula for the x - coordinate of a point \(N(x,y)\) that divides the line segment joining \(L(x_1,y_1)\) and \(M(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\). Here, \(x_1=-7\), \(x_2 = 5\), \(m = 1\) and \(n = 3\). So, \(x=\frac{1\times5+3\times(-7)}{1 + 3}\).
\[x=\frac{5-21}{4}=\frac{-16}{4}=-4\]

Step2: Use the section - formula for y - coordinate

The section formula for the y - coordinate of a point \(N(x,y)\) that divides the line segment joining \(L(x_1,y_1)\) and \(M(x_2,y_2)\) in the ratio \(m:n\) is \(y=\frac{my_2+ny_1}{m + n}\). Here, \(y_1=-7\), \(y_2 = 9\), \(m = 1\) and \(n = 3\). So, \(y=\frac{1\times9+3\times(-7)}{1 + 3}\).
\[y=\frac{9-21}{4}=\frac{-12}{4}=-3\]

Answer:

\((-4,-3)\)