Question

what are the coordinates of the point on the directed line segment from (-7, 9) to (3, -1) that partitions the segment into a ratio of 2 to 3?

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, \((x_1,y_1)=(-7,9)\), \((x_2,y_2)=(3,-1)\), \(m = 2\), and \(n = 3\).

Step2: Calculate the x - coordinate

\[

$$\begin{align*} x&=\frac{2\times3+3\times(-7)}{2 + 3}\\ &=\frac{6-21}{5}\\ &=\frac{-15}{5}\\ &=-3 \end{align*}$$

\]

Step3: Calculate the y - coordinate

\[

$$\begin{align*} y&=\frac{2\times(-1)+3\times9}{2 + 3}\\ &=\frac{-2 + 27}{5}\\ &=\frac{25}{5}\\ &=5 \end{align*}$$

\]

Answer:

\((-3,5)\)