Question

the coordinates of the endpoints of $overline{no}$ are $n(5,7)$ and $o(13,15)$. point $p$ is on $overline{no}$ and divides it such that $np:op$ is $3:1$. what are the coordinates of $p$? 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 \(P(x,y)\) that divides the line - segment joining \(N(x_1,y_1)\) and \(O(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\). Here, \(x_1 = 5\), \(x_2=13\), \(m = 3\), and \(n = 1\). So, \(x=\frac{3\times13+1\times5}{3 + 1}=\frac{39 + 5}{4}=\frac{44}{4}=11\).

Step2: Use the section - formula for y - coordinate

The section formula for the y - coordinate of a point \(P(x,y)\) that divides the line - segment joining \(N(x_1,y_1)\) and \(O(x_2,y_2)\) in the ratio \(m:n\) is \(y=\frac{my_2+ny_1}{m + n}\). Here, \(y_1 = 7\), \(y_2=15\), \(m = 3\), and \(n = 1\). So, \(y=\frac{3\times15+1\times7}{3 + 1}=\frac{45+7}{4}=\frac{52}{4}=13\).

Answer:

\((11,13)\)