Question

what are the coordinates of the point on the directed line segment from (-10,5) to (8,2) that partitions the segment into a ratio of 1 to 2?

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, \((x_1,y_1)=(-10,5)\), \((x_2,y_2)=(8,2)\), \(m = 1\), and \(n = 2\).

Step2: Calculate the x - coordinate

Substitute the values into the x - coordinate formula:
\[

$$\begin{align*} x&=\frac{1\times8+2\times(-10)}{1 + 2}\\ &=\frac{8-20}{3}\\ &=\frac{-12}{3}\\ &=-4 \end{align*}$$

\]

Step3: Calculate the y - coordinate

Substitute the values into the y - coordinate formula:
\[

$$\begin{align*} y&=\frac{1\times2+2\times5}{1 + 2}\\ &=\frac{2 + 10}{3}\\ &=\frac{12}{3}\\ &=4 \end{align*}$$

\]

Answer:

\((-4,4)\)