Question

what are the coordinates of the point on the directed line segment from (-1, -9) to (8, 9) that partitions the segment into a ratio of 4 to 5?

Explanation:

Step1: Recall the section - formula

For a line - segment with endpoints $(x_1,y_1)$ and $(x_2,y_2)$ partitioned in the ratio $m:n$, the coordinates of the point $P(x,y)$ are given by $x=\frac{mx_2+nx_1}{m + n}$ and $y=\frac{my_2+ny_1}{m + n}$. Here, $(x_1,y_1)=(-1,-9)$, $(x_2,y_2)=(8,9)$, $m = 4$, and $n = 5$.

Step2: Calculate the x - coordinate

$x=\frac{4\times8+5\times(-1)}{4 + 5}=\frac{32-5}{9}=\frac{27}{9}=3$.

Step3: Calculate the y - coordinate

$y=\frac{4\times9+5\times(-9)}{4 + 5}=\frac{36-45}{9}=\frac{-9}{9}=-1$.

Answer:

$(3,-1)$