Question

what are the coordinates of the point on the directed line segment from (-7,2) to (-4,-10) that partitions the segment into a ratio of 5 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, \((x_1,y_1)=(-7,2)\), \((x_2,y_2)=(-4,-10)\), \(m = 5\), and \(n = 1\).

Step2: Calculate the \(x\) - coordinate

\[

$$\begin{align*} x&=\frac{5\times(-4)+1\times(-7)}{5 + 1}\\ &=\frac{-20-7}{6}\\ &=\frac{-27}{6}\\ &=-\frac{9}{2}=-4.5 \end{align*}$$

\]

Step3: Calculate the \(y\) - coordinate

\[

$$\begin{align*} y&=\frac{5\times(-10)+1\times2}{5 + 1}\\ &=\frac{-50 + 2}{6}\\ &=\frac{-48}{6}\\ &=-8 \end{align*}$$

\]

Answer:

\((-4.5,-8)\)