Question

in the diagram below, $overline{ac}$ has endpoints with coordinates $a(-5,2)$ and $c(4,-10)$. if $b$ is a point on $overline{ac}$ and $ab:bc = 1:2$, what are the coordinates of $b$? step 1: label $(x_1,y_1)$ and $(x_2,y_2)$ on the given coordinates step 2: find the value of $k = a/(a + b)$ step 3: substitute x and y coordinates to find the partitioned coordinates

Explanation:

Step1: Identify the values of $(x_1,y_1)$ and $(x_2,y_2)$

Let $A(-5,2)$ be $(x_1,y_1)$ and $C(4,-10)$ be $(x_2,y_2)$. Also, since $AB:BC = 1:2$, then $a = 1$ and $b=2$, and $k=\frac{a}{a + b}=\frac{1}{1 + 2}=\frac{1}{3}$.

Step2: Calculate the x - coordinate of point B

The formula for the x - coordinate of the point that divides a line - segment in the ratio $a:b$ is $x=x_1 + k(x_2 - x_1)$.
Substitute $x_1=-5$, $x_2 = 4$, and $k=\frac{1}{3}$ into the formula:
$x=-5+\frac{1}{3}(4-(-5))=-5+\frac{1}{3}(4 + 5)=-5 + 3=-2$.

Step3: Calculate the y - coordinate of point B

The formula for the y - coordinate of the point that divides a line - segment in the ratio $a:b$ is $y=y_1 + k(y_2 - y_1)$.
Substitute $y_1 = 2$, $y_2=-10$, and $k=\frac{1}{3}$ into the formula:
$y=2+\frac{1}{3}(-10 - 2)=2+\frac{1}{3}(-12)=2-4=-2$.

Answer:

$(-2,-2)$