Question

what are the coordinates of the point on the directed line segment from (-1,6) to (3,-2) that partitions the segment into a ratio of 5 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 \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(x_1=-1,y_1 = 6,x_2 = 3,y_2=-2,m = 5,n = 3\).

Step2: Calculate the \(x\) - coordinate

\[

$$\begin{align*} x&=\frac{5\times3+3\times(-1)}{5 + 3}\\ &=\frac{15-3}{8}\\ &=\frac{12}{8}\\ &=\frac{3}{2} \end{align*}$$

\]

Step3: Calculate the \(y\) - coordinate

\[

$$\begin{align*} y&=\frac{5\times(-2)+3\times6}{5 + 3}\\ &=\frac{-10 + 18}{8}\\ &=\frac{8}{8}\\ &=1 \end{align*}$$

\]

Answer:

\((\frac{3}{2},1)\)