Question

calculate point p on \overline{ab} such that the ratio of ap to pb is 1:3

Explanation:

Step1: Find the coordinates of \(A\) and \(B\)

From the graph, \(A(-3,-4)\) and \(B(2,3)\)

Step2: Use 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 \(m = 1\) and \(n = 3\), \(x_1=-3,y_1=-4,x_2 = 2,y_2=3\)

For \(x\)-coordinate:

\(x=\frac{1\times2+3\times(-3)}{1 + 3}=\frac{2-9}{4}=\frac{-7}{4}=-1.75\)

For \(y\)-coordinate:

\(y=\frac{1\times3+3\times(-4)}{1 + 3}=\frac{3-12}{4}=\frac{-9}{4}=-2.25\)

Answer:

The coordinates of point \(P\) are \((-1.75,-2.25)\)