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: Assume coordinates of \(A\) and \(B\)

Let \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\). From the graph, assume \(A=(- 4, - 1)\) and \(B=(5, - 1)\).

Step3: Calculate the \(x\) - coordinate of \(P\)

Using the formula \(x=\frac{mx_2+nx_1}{m + n}\), substitute \(m = 2\), \(n = 1\), \(x_1=-4\), and \(x_2 = 5\).
\[x=\frac{2\times5+1\times(-4)}{2 + 1}=\frac{10 - 4}{3}=\frac{6}{3}=2\]

Step4: Calculate the \(y\) - coordinate of \(P\)

Using the formula \(y=\frac{my_2+ny_1}{m + n}\), substitute \(m = 2\), \(n = 1\), \(y_1=-1\), and \(y_2=-1\).
\[y=\frac{2\times(-1)+1\times(-1)}{2 + 1}=\frac{-2-1}{3}=\frac{-3}{3}=-1\]

Answer:

The coordinates of point \(P\) are \((2,-1)\)