Question

what are the coordinates of the point on the directed line segment from (1,5) to (6,10) that partitions the segment into a ratio of 3 to 2? answer attempt 1 out of 2

Explanation:

Step1: Recall the section - formula

The formula for finding 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, \((x_1,y_1)=(1,5)\), \((x_2,y_2)=(6,10)\), \(m = 3\), and \(n = 2\).

Step2: Calculate the x - coordinate

\[

$$\begin{align*} x&=\frac{3\times6+2\times1}{3 + 2}\\ &=\frac{18 + 2}{5}\\ &=\frac{20}{5}\\ &=4 \end{align*}$$

\]

Step3: Calculate the y - coordinate

\[

$$\begin{align*} y&=\frac{3\times10+2\times5}{3 + 2}\\ &=\frac{30+10}{5}\\ &=\frac{40}{5}\\ &=8 \end{align*}$$

\]

Answer:

\((4,8)\)