Question

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 $J(1,1)$ and its image $J'(5,1)$.

Step2: Calculate scale - factor

The distance from the center of dilation to a point and its image is related by the scale - factor. Since the $y$ - coordinates of $J$ and $J'$ are the same, we can look at the $x$ - coordinates. The distance from the center of dilation (which we will find is the origin $(0,0)$) to $J$ in the $x$ - direction is $1 - 0=1$, and the distance from the center of dilation to $J'$ in the $x$ - direction is $5 - 0 = 5$. The scale factor $k=\frac{\text{distance from center to image}}{\text{distance from center to pre - image}}$. So, $k = 5$.

Step3: Find the center of dilation

We can use the property of dilation. If we consider another pair of points, say $K(-2,-2)$ and $K'( - 10,-10)$. Let the center of dilation be $(x_0,y_0)$. The formula for dilation is $(x,y)\to(x_0 + k(x - x_0),y_0 + k(y - y_0))$. If we assume the center of dilation is the origin $(0,0)$, for point $K(-2,-2)$ with $k = 5$, the image is $(0+5(-2 - 0),0 + 5(-2 - 0))=(-10,-10)$. So the center of dilation is $(0,0)$.

Answer:

scale factor: $5$
center of the dilation: $(0,0)$