Question

1. given directed line segment ab, find the coordinates of p such that the ratio of ap to pb is 2:1. plot point p.

Explanation:

Step1: Recall the section - formula

If a point $P(x,y)$ divides the line - segment joining $A(x_1,y_1)$ and $B(x_2,y_2)$ in the ratio $m:n$, then the coordinates of $P$ are given by $x=\frac{mx_2+nx_1}{m + n}$ and $y=\frac{my_2+ny_1}{m + n}$. Here, $m = 2$ and $n = 1$.

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

Let $A(x_1,y_1)$ and $B(x_2,y_2)$. Then $x=\frac{2x_2+1x_1}{2 + 1}=\frac{2x_2+x_1}{3}$.

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

$y=\frac{2y_2+1y_1}{2 + 1}=\frac{2y_2+y_1}{3}$.

Answer:

The coordinates of $P$ are $(\frac{2x_2+x_1}{3},\frac{2y_2+y_1}{3})$