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. 1 - 3 partitioning a segment homework name:

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 \(x=\frac{mx_2+nx_1}{m + n}\) 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 \(P\)

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

Since the coordinates of \(A(x_1,y_1)\) and \(B(x_2,y_2)\) are not given in the problem statement, the general coordinates of \(P\) that divides the line - segment \(\overline{AB}\) in the ratio \(2:1\) are \((\frac{2x_2+x_1}{3},\frac{2y_2+y_1}{3})\).

Answer:

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