Question

what are the coordinates of the point on the directed line segment from (-7,6) to (7,-2) that partitions the segment into a ratio of 1 to 3?

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 given by \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(x_1=-7,y_1 = 6,x_2 = 7,y_2=-2,m = 1,n = 3\).

Step2: Calculate the \(x\) - coordinate

Substitute the values into the \(x\) - coordinate formula:

$$x=\frac{1\times7+3\times(-7)}{1 + 3}=\frac{7-21}{4}=\frac{-14}{4}=-\frac{7}{2}$$

Step3: Calculate the \(y\) - coordinate

Substitute the values into the \(y\) - coordinate formula:

$$y=\frac{1\times(-2)+3\times6}{1 + 3}=\frac{-2 + 18}{4}=\frac{16}{4}=4$$

Answer:

\((-\frac{7}{2},4)\)