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the coordinates of the endpoints of directed line segment abc are ( a(-8,7) ) and ( c(7,-13) ). if ( ab:bc = 3:2 ), the coordinates of b are
a) ( (1,-5) )
b) ( (-2,-1) )
c) ( (-3,0) )
d) ( (3,-6) )

Question

Accepted Answer

# Explanation:
## Step1: Use the 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 $m:n$, then 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, $m = 3$, $n=2$, $x_1=-8$, $y_1 = 7$, $x_2=7$, $y_2=-13$.
## Step2: Calculate the $x$ - coordinate of $B$
Substitute the values into the $x$ - coordinate formula:
$$
\begin{align*}
x&=\frac{3\times7+2\times(-8)}{3 + 2}\
&=\frac{21-16}{5}\
&=\frac{5}{5}\
&=1
\end{align*}
$$
## Step3: Calculate the $y$ - coordinate of $B$
Substitute the values into the $y$ - coordinate formula:
$$
\begin{align*}
y&=\frac{3\times(-13)+2\times7}{3 + 2}\
&=\frac{-39 + 14}{5}\
&=\frac{-25}{5}\
&=-5
\end{align*}
$$

# Answer:
A. $(1,-5)$

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QUESTION IMAGE

Question

Explanation:

Step1: Use the section formula

Step2: Calculate the \(x\) - coordinate of \(B\)

Step3: Calculate the \(y\) - coordinate of \(B\)

Answer: