Question

what are the coordinates of the point on the directed line segment from (-6,4) to (-3,10) that partitions the segment into a ratio of 1 to 2?

Explanation:

Step1: Recall the section formula

The section formula for a point \((x,y)\) that divides the line segment joining \((x_1,y_1)\) and \((x_2,y_2)\) 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, \((x_1,y_1)=(-6,4)\), \((x_2,y_2)=(-3,10)\), \(m = 1\) and \(n=2\).

Step2: Calculate the x - coordinate

Substitute the values into the formula for \(x\):
\(x=\frac{1\times(-3)+2\times(-6)}{1 + 2}=\frac{-3-12}{3}=\frac{-15}{3}=-5\)

Step3: Calculate the y - coordinate

Substitute the values into the formula for \(y\):
\(y=\frac{1\times10+2\times4}{1 + 2}=\frac{10 + 8}{3}=\frac{18}{3}=6\)

Answer:

The coordinates of the point are \((-5,6)\)