Question

find the point that divides the directed line segment from (10, -10) to (-5, -25) in a 2:1 ratio.

Explanation:

Step1: Recall the section formula

The section formula for a point \((x,y)\) that divides the line segment joining \((x_1,y_1)\) and \((x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2 + nx_1}{m + n}\) and \(y=\frac{my_2 + ny_1}{m + n}\). Here, \((x_1,y_1)=(10,-10)\), \((x_2,y_2)=(-5,-25)\), and \(m:n = 2:1\), so \(m = 2\), \(n = 1\).

Step2: Calculate the x - coordinate

Substitute the values into the x - coordinate formula: \(x=\frac{2\times(-5)+1\times10}{2 + 1}=\frac{-10 + 10}{3}=\frac{0}{3}=0\).

Step3: Calculate the y - coordinate

Substitute the values into the y - coordinate formula: \(y=\frac{2\times(-25)+1\times(-10)}{2 + 1}=\frac{-50-10}{3}=\frac{-60}{3}=-20\).

Answer:

The point is \((0,-20)\)