Question

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

Explanation:

Step1: Recall the section - formula

The formula to find the coordinates of 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 $x=\frac{mx_2+nx_1}{m + n}$ and $y=\frac{my_2+ny_1}{m + n}$. Here, $(x_1,y_1)=(-6,-6)$, $(x_2,y_2)=(1,1)$, $m = 1$, and $n = 6$.

Step2: Calculate the x - coordinate

Substitute the values into the x - coordinate formula:
$x=\frac{1\times1+6\times(-6)}{1 + 6}=\frac{1-36}{7}=\frac{-35}{7}=-5$.

Step3: Calculate the y - coordinate

Substitute the values into the y - coordinate formula:
$y=\frac{1\times1+6\times(-6)}{1 + 6}=\frac{1 - 36}{7}=\frac{-35}{7}=-5$.

Answer:

$(-5,-5)$