Question

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point a is at (-6, -5) and point c is at (4, 0).
find the coordinates of point b on $overline{ac}$ such that the ratio of ab to bc is 2 : 3.
b = ( , )

Explanation:

Step1: Use the section - formula for x - coordinate

The 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, \(x_1=-6\), \(x_2 = 4\), \(m = 2\), \(n=3\).

$$x=\frac{2\times4+3\times(-6)}{2 + 3}=\frac{8-18}{5}=\frac{-10}{5}=-2$$

Step2: Use the section - formula for y - coordinate

The 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, \(y_1=-5\), \(y_2 = 0\), \(m = 2\), \(n=3\).

$$y=\frac{2\times0+3\times(-5)}{2 + 3}=\frac{0 - 15}{5}=-3$$

Answer:

\((-2,-3)\)