Question

figure def has coordinates d(10,5), e(3,0), and f(20,8) and is dilated to create a new figure with coordinates d(2,1), e(0.6,0), and f(4,1.6). 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

To find the scale factor $k$, we can use the ratio of the coordinates of a corresponding point. Let's use point $D(10,5)$ and $D'(2,1)$. For the $x$ - coordinates, $k=\frac{x_{D'}}{x_{D}}=\frac{2}{10} = 0.2$, and for the $y$ - coordinates, $k=\frac{y_{D'}}{y_{D}}=\frac{1}{5}=0.2$. So the scale factor $k = 0.2$.

Step2: Determine the rule

The rule for dilation with a scale factor $k$ centered at the origin $(0,0)$ is $(x,y)\to(kx,ky)$. Since $k = 0.2$, the rule is $(x,y)\to(0.2x,0.2y)$.

Step3: Find the center of dilation

Since the ratios of the coordinates of all corresponding points are the same and there is no translation involved in the transformation (the transformation is a simple scaling), the center of dilation is the origin $(0,0)$.

Answer:

Scale Factor: $0.2$
Rule: $(x,y)\to(0.2x,0.2y)$
Center of Dilation: $(0,0)$