Question

what are the coordinates of the point on the directed line segment from (-6,4) to (8,-10) that partitions the segment into a ratio of 1 to 6?

Explanation:

Step1: Recall the section - formula

The formula for finding 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, \(x_1=-6,y_1 = 4,x_2 = 8,y_2=-10,m = 1,n = 6\).

Step2: Calculate the x - coordinate

\[

$$\begin{align*} x&=\frac{1\times8+6\times(-6)}{1 + 6}\\ &=\frac{8-36}{7}\\ &=\frac{-28}{7}\\ &=-4 \end{align*}$$

\]

Step3: Calculate the y - coordinate

\[

$$\begin{align*} y&=\frac{1\times(-10)+6\times4}{1 + 6}\\ &=\frac{-10 + 24}{7}\\ &=\frac{14}{7}\\ &=2 \end{align*}$$

\]

Answer:

\((-4,2)\)