Question

what are the coordinates of the point on the directed line segment from (-10, 7) to (4, -9) that partitions the segment into a ratio of 3 to 1?

Explanation:

Step1: Recall the section - formula

The formula to find the coordinates of a point \(P(x,y)\) 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, \((x_1,y_1)=(-10,7)\), \((x_2,y_2)=(4,-9)\), \(m = 3\), and \(n = 1\).

Step2: Calculate the x - coordinate

\[

$$\begin{align*} x&=\frac{3\times4+1\times(-10)}{3 + 1}\\ &=\frac{12-10}{4}\\ &=\frac{2}{4}\\ &=\frac{1}{2} \end{align*}$$

\]

Step3: Calculate the y - coordinate

\[

$$\begin{align*} y&=\frac{3\times(-9)+1\times7}{3 + 1}\\ &=\frac{-27 + 7}{4}\\ &=\frac{-20}{4}\\ &=-5 \end{align*}$$

\]

Answer:

\((\frac{1}{2},-5)\)