Question

point x on $overline{jk}$ is $\frac{1}{3}$ of the distance from j to k. j(-1,4) k(5, -3)

Explanation:

Step1: Find the coordinates of endpoints

The endpoints of the line - segment are \(J(-1,4)\) and \(K(5, - 3)\).

Step2: Use the section - formula

If a point \(X\) divides the line - segment joining \(J(x_1,y_1)\) and \(K(x_2,y_2)\) in the ratio \(m:n\), and here \(m = 1\) and \(n=3 - 1=2\). The section - formula for the \(x\) - coordinate of \(X\) is \(x=\frac{mx_2+nx_1}{m + n}\) and for the \(y\) - coordinate is \(y=\frac{my_2+ny_1}{m + n}\).
For \(x\) - coordinate:
\[

$$\begin{align*} x&=\frac{1\times5+2\times(-1)}{1 + 2}\\ &=\frac{5-2}{3}\\ &=1 \end{align*}$$

\]
For \(y\) - coordinate:
\[

$$\begin{align*} y&=\frac{1\times(-3)+2\times4}{1 + 2}\\ &=\frac{-3 + 8}{3}\\ &=\frac{5}{3} \end{align*}$$

\]

Answer:

The coordinates of point \(X\) are \((1,\frac{5}{3})\)