Question

what are the coordinates of the point on the directed line segment from (-10,9) to (-4,3) that partitions the segment into a ratio of 1 to 5?

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, \((x_1,y_1)=(-10,9)\), \((x_2,y_2)=(-4,3)\), \(m = 1\), and \(n = 5\).

Step2: Calculate the \(x\) - coordinate

\[

$$\begin{align*} x&=\frac{1\times(-4)+5\times(-10)}{1 + 5}\\ &=\frac{-4-50}{6}\\ &=\frac{-54}{6}\\ &=-9 \end{align*}$$

\]

Step3: Calculate the \(y\) - coordinate

\[

$$\begin{align*} y&=\frac{1\times3+5\times9}{1 + 5}\\ &=\frac{3 + 45}{6}\\ &=\frac{48}{6}\\ &=8 \end{align*}$$

\]

Answer:

\((-9,8)\)