Question

the point b lies on the segment $overline{ac}$. find the coordinates of b so that ab is $\frac{1}{6}$ of ac. c (3, 1) b (?,?) a (-21, -17) coordinates of b :

Explanation:

Step1: Use section - formula

If a point $B(x,y)$ divides the line - segment joining $A(x_1,y_1)$ and $C(x_2,y_2)$ in the ratio $m:n$, the coordinates of $B$ are given by $x=\frac{mx_2+nx_1}{m + n}$ and $y=\frac{my_2+ny_1}{m + n}$. Here, $m = 1$ and $n=5$ (since $AB:BC=1:5$ as $AB=\frac{1}{6}AC$), $x_1=-21$, $y_1=-17$, $x_2 = 3$, and $y_2 = 1$.

Step2: Calculate the x - coordinate of B

$x=\frac{1\times3+5\times(-21)}{1 + 5}=\frac{3-105}{6}=\frac{-102}{6}=-17$.

Step3: Calculate the y - coordinate of B

$y=\frac{1\times1+5\times(-17)}{1 + 5}=\frac{1-85}{6}=\frac{-84}{6}=-14$.

Answer:

$(-17,-14)$