Question

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

Explanation:

Step1: Recall the section - formula

The formula to find 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, \((x_1,y_1)=(2,-6)\), \((x_2,y_2)=(5,6)\), \(m = 1\), and \(n = 5\).

Step2: Calculate the \(x\) - coordinate

Substitute the values into the \(x\) - coordinate formula:
\[x=\frac{1\times5+5\times2}{1 + 5}=\frac{5 + 10}{6}=\frac{15}{6}=\frac{5}{2}\]

Step3: Calculate the \(y\) - coordinate

Substitute the values into the \(y\) - coordinate formula:
\[y=\frac{1\times6+5\times(-6)}{1 + 5}=\frac{6-30}{6}=\frac{-24}{6}=-4\]

Answer:

\((\frac{5}{2},-4)\)