Question

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

Explanation:

1. First, assume the coordinates of point \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\). From the graph (not shown in detail here for generality), if we use the section - formula for a point \(P=(x,y)\) that divides the line - segment joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\), the formulas are:

• \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(m = 2\) and \(n = 1\).

1. Let's assume \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\).

• Calculate the \(x\) - coordinate of \(P\):
• \(x=\frac{2x_2 + 1x_1}{2 + 1}=\frac{2x_2+x_1}{3}\).
• Calculate the \(y\) - coordinate of \(P\):
• \(y=\frac{2y_2 + 1y_1}{2 + 1}=\frac{2y_2+y_1}{3}\).
• For example, if \(A=(x_1,y_1)=( - 2,-3)\) and \(B=(x_2,y_2)=(1,3)\):
• \(x=\frac{2\times1+1\times(-2)}{3}=\frac{2 - 2}{3}=0\).
• \(y=\frac{2\times3+1\times(-3)}{3}=\frac{6 - 3}{3}=1\). So \(P=(0,1)\).

Answer:

The coordinates of \(P\) are \((\frac{2x_2+x_1}{3},\frac{2y_2+y_1}{3})\). To plot \(P\), first find the \(x\) and \(y\) values using the above - mentioned formula with the actual coordinates of \(A\) and \(B\) from the graph, and then mark the point on the coordinate plane.