Question

what are the coordinates of the point on the directed line segment from (-9, -10) to (1, 5) that partitions the segment into a ratio of 2 to 3?

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:
\[x=\frac{mx_2 + nx_1}{m + n}, \quad y=\frac{my_2 + ny_1}{m + n}\]
Here, \((x_1,y_1)=(-9,-10)\), \((x_2,y_2)=(1,5)\), \(m = 2\) and \(n=3\).

Step2: Calculate the x - coordinate

Substitute the values into the formula for \(x\):
\[x=\frac{2\times1+3\times(-9)}{2 + 3}=\frac{2-27}{5}=\frac{-25}{5}=-5\]

Step3: Calculate the y - coordinate

Substitute the values into the formula for \(y\):
\[y=\frac{2\times5+3\times(-10)}{2 + 3}=\frac{10-30}{5}=\frac{-20}{5}=-4\]

Answer:

The coordinates of the point are \((-5,-4)\)