Question

point b partitions $overline{ac}$ in the ratio 1:3. what are the coordinates of c? the coordinates of c are (simplify your answer. type an ordered pair.)

Explanation:

Step1: Recall 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 $x=\frac{mx_2+nx_1}{m + n}$ and $y=\frac{my_2+ny_1}{m + n}$. Here $m = 1$ and $n = 3$. Let the coordinates of $A$ be $(x_1,y_1)$ and of $B$ be $(x,y)$ and of $C$ be $(x_2,y_2)$. We can re - arrange the formula for $x_2$ and $y_2$: $x_2=\frac{(m + n)x−nx_1}{m}$ and $y_2=\frac{(m + n)y−ny_1}{m}$.

Step2: Assume coordinates from the graph

Suppose the coordinates of $A$ are $(-4,-5)$ and the coordinates of $B$ are $(3,0)$.

Step3: Calculate the $x$ - coordinate of $C$

Using the formula $x_2=\frac{(m + n)x−nx_1}{m}$, substituting $m = 1$, $n = 3$, $x_1=-4$, and $x = 3$.
\[

$$\begin{align*} x_2&=\frac{(1 + 3)\times3-3\times(-4)}{1}\\ &=\frac{4\times3+12}{1}\\ &=\frac{12 + 12}{1}\\ &=24 \end{align*}$$

\]

Step4: Calculate the $y$ - coordinate of $C$

Using the formula $y_2=\frac{(m + n)y−ny_1}{m}$, substituting $m = 1$, $n = 3$, $y_1=-5$, and $y = 0$.
\[

$$\begin{align*} y_2&=\frac{(1 + 3)\times0-3\times(-5)}{1}\\ &=\frac{0 + 15}{1}\\ &=15 \end{align*}$$

\]

Answer:

$(24,15)$