Question

what are the coordinates of the point on the directed line segment from (-6,4) to (-2,-4) that partitions the segment into a ratio of 3 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, \(x_1=-6,y_1 = 4,x_2=-2,y_2=-4,m = 3,n = 5\).

Step2: Calculate the x - coordinate

\[

$$\begin{align*} x&=\frac{3\times(-2)+5\times(-6)}{3 + 5}\\ &=\frac{-6-30}{8}\\ &=\frac{-36}{8}\\ &=-\frac{9}{2}=-4.5 \end{align*}$$

\]

Step3: Calculate the y - coordinate

\[

$$\begin{align*} y&=\frac{3\times(-4)+5\times4}{3 + 5}\\ &=\frac{-12 + 20}{8}\\ &=\frac{8}{8}\\ &=1 \end{align*}$$

\]

Answer:

\((-4.5,1)\)