Question

in the graph below, rhombus jklm is the image of jklm 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: Recall dilation properties

Dilation is a transformation that changes the size of a figure. To find the scale - factor, we can compare the lengths of corresponding sides of the pre - image and the image. Let's consider the distance between two points on the pre - image and the image. For example, let's take point $K(2,-2)$ and its image $K'(2,2)$.

Step2: Calculate the scale factor

The distance of point $K$ from the $x$ - axis is $2$ units (in the negative $y$ direction) and the distance of $K'$ from the $x$ - axis is $2$ units (in the positive $y$ direction). The ratio of the distance of the image point from the reference (in this case, the $x$ - axis) to the distance of the pre - image point from the reference gives the scale factor. Since the distance of $K'$ from the $x$ - axis is in the opposite direction of $K$ from the $x$ - axis and their magnitudes are the same, the scale factor $k=- 1$.

Step3: Find the center of dilation

The center of dilation is the point that remains fixed during the dilation. We can observe that the $x$ - coordinate of corresponding points (e.g., $K$ and $K'$, $L$ and $L'$) remains the same. Also, the mid - point between a point and its image lies on the center of dilation. For points $K(2,-2)$ and $K'(2,2)$, the mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Plugging in $x_1 = 2,y_1=-2,x_2 = 2,y_2 = 2$, we get $(\frac{2 + 2}{2},\frac{-2+2}{2})=(2,0)$.

Answer:

scale factor: $-1$
center of the dilation: $(2,0)$