Question

what are the coordinates of the point on the directed line segment from (-5, -10) to (2, 4) that partitions the segment into a ratio of 2 to 5?

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, \(x_1=-5,y_1=-10,x_2 = 2,y_2 = 4,m = 2,n = 5\).

Step2: Calculate the x - coordinate

\[

$$\begin{align*} x&=\frac{2\times2+5\times(-5)}{2 + 5}\\ &=\frac{4-25}{7}\\ &=\frac{-21}{7}\\ &=-3 \end{align*}$$

\]

Step3: Calculate the y - coordinate

\[

$$\begin{align*} y&=\frac{2\times4+5\times(-10)}{2 + 5}\\ &=\frac{8-50}{7}\\ &=\frac{-42}{7}\\ &=-6 \end{align*}$$

\]

Answer:

\((-3,-6)\)