Question

figure rst has coordinates r(10, 10), s(15, 6), and t(20, 10) and is dilated to create a new figure with coordinates r(25, 25), s(37.5, 15), and t(50, 25). determine the scale factor, the center of dilation, and enter the rule applied to create the new figure. > enter the answer in each space provided. use numbers instead of words. scale factor rule (x, y)→( , ) center of dilation ( , )

Explanation:

Step1: Calculate scale factor

For point $R(10,10)$ and $R'(25,25)$, divide the coordinates of $R'$ by coordinates of $R$. $\frac{25}{10}=\frac{5}{2} = 2.5$. Checking with other points: for $S(15,6)$ and $S'(37.5,15)$, $\frac{37.5}{15}=2.5$ and $\frac{15}{6}=2.5$. So the scale factor $k = 2.5$.

Step2: Find center of dilation

Let the center of dilation be $(a,b)$. The formula for dilation is $(x',y')=k(x - a)+a,k(y - b)+b$. Using point $R(10,10)$ and $R'(25,25)$ and $k = 2.5$, we have the system of equations:
$25=2.5(10 - a)+a$ and $25=2.5(10 - b)+b$.
For the $x$ - coordinate:
\[

$$\begin{align*} 25&=25-2.5a+a\\ 25&=25 - 1.5a\\ 1.5a&=0\\ a&=0 \end{align*}$$

\]
For the $y$ - coordinate:
\[

$$\begin{align*} 25&=25-2.5b + b\\ 25&=25-1.5b\\ 1.5b&=0\\ b&=0 \end{align*}$$

\]
So the center of dilation is $(0,0)$.

Step3: Determine the rule

The rule for dilation with scale factor $k = 2.5$ and center of dilation $(0,0)$ is $(x,y)\to(2.5x,2.5y)$.

Answer:

Scale Factor: $2.5$
Rule: $(x,y)\to(2.5x,2.5y)$
Center of Dilation: $(0,0)$