Question

in the graph below, hexagon ghijkl is the image of ghijkl after a dilation. what are the scale factor and center of the dilation? simplify your answers and write them as fractions or whole numbers. scale factor: center of the dilation: ( , )

Explanation:

Step1: Identify corresponding points

Let's take point $G(-5,5)$ and its image $G'(-10,10)$.

Step2: Calculate the scale - factor

The scale - factor $k$ can be found by looking at the ratio of the coordinates of the image to the pre - image. For the $x$ - coordinates: $\frac{x_{G'}}{x_{G}}=\frac{- 10}{-5}=2$, and for the $y$ - coordinates: $\frac{y_{G'}}{y_{G}}=\frac{10}{5}=2$. So the scale factor $k = 2$.

Step3: Find the center of dilation

The center of dilation is the point that all the points and their images are dilated from. By observing the graph, we can see that the center of dilation is the origin $(0,0)$. We can also verify this by using the property of dilation. If the center of dilation is $(a,b)$ and the scale factor is $k$, then for a point $(x,y)$ and its image $(x',y')$ the formulas are $x'=a + k(x - a)$ and $y'=b + k(y - b)$. When $(a,b)=(0,0)$, $x'=kx$ and $y'=ky$ which holds for all corresponding points in the dilation.

Answer:

scale factor: $2$
center of the dilation: $(0,0)$