Question

the point b lies on the segment (overline{ac}). find the coordinates of b so that the ratio of ab to bc is 1 to 2. c (9,13) b (?,?) a (-3, -2)

Explanation:

Step1: Use the section - formula for x - coordinate

The section - formula for the x - coordinate of a point \(B(x,y)\) that divides the line segment joining \(A(x_1,y_1)\) and \(C(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\). Here, \(x_1=-3\), \(x_2 = 9\), \(m = 1\), and \(n = 2\).
\[x=\frac{1\times9+2\times(-3)}{1 + 2}=\frac{9-6}{3}=\frac{3}{3}=1\]

Step2: Use the section - formula for y - coordinate

The section - formula for the y - coordinate of a point \(B(x,y)\) that divides the line segment joining \(A(x_1,y_1)\) and \(C(x_2,y_2)\) in the ratio \(m:n\) is \(y=\frac{my_2+ny_1}{m + n}\). Here, \(y_1=-2\), \(y_2 = 13\), \(m = 1\), and \(n = 2\).
\[y=\frac{1\times13+2\times(-2)}{1 + 2}=\frac{13 - 4}{3}=\frac{9}{3}=3\]

Answer:

\((1,3)\)