Question

triangle efg is the image that resulted from a dilation of △abc. determine the center of dilation and the scale factor of the transformation. then give the rule applied to create the new figure. > enter the answer in each space provided. use numbers instead of words. △abc was dilated by a scale factor of using ( , ) as the center of dilation. rule: (x,y)→( , )

Explanation:

Step1: Identify corresponding points

Let's take point $A(- 2,1)$ and its corresponding point $E(-8,1)$. Also, consider $B(1,3)$ and $F(2,7)$ and $C(2,1)$ and $G(4,1)$.

Step2: Find the center of dilation

By observing the graph, we can see that the center of dilation is the origin $(0,0)$ since the lines connecting corresponding points (e.g., $AE$, $BF$, $CG$) pass through the origin.

Step3: Calculate the scale - factor

For the $x$ - coordinates, if we consider point $A(-2,1)$ and $E(-8,1)$, the ratio of the $x$ - coordinates of the image to the pre - image is $\frac{-8}{-2}=4$. For the $y$ - coordinates, considering $B(1,3)$ and $F(2,7)$, the vertical distance from the center of dilation to $B$ in the $y$ - direction is $3$ and to $F$ is $7$. The ratio of the $y$ - coordinates of the image to the pre - image is also $4$. So the scale factor $k = 4$.

Step4: Write the dilation rule

The rule for a dilation with center $(0,0)$ and scale factor $k$ is $(x,y)\to(kx,ky)$. Since $k = 4$, the rule is $(x,y)\to(4x,4y)$.

Answer:

$\Delta ABC$ was dilated by a scale factor of $4$ using $(0,0)$ as the center of dilation.
Rule: $(x,y)\to(4x,4y)$