Question

in the graph below, △jkl is the image of △jkl 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 $L(-2,3)$ and its image $L'(1,3)$.

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 $L$ and $L'$ are the same, we can consider the change in the $x$ - coordinates. The distance from the center of dilation (which we will find later) to $L$ and $L'$ in the $x$ - direction. The distance between the $x$ - coordinates of $L$ and $L'$ is $1-(-2)=3$. Let's assume the center of dilation is on the line $y = 3$ (because the $y$ - coordinates of corresponding points on $LL'$ are the same). Let's take another pair of corresponding points, say $K(-7,3)$ and $K'(-4,3)$. The distance between their $x$ - coordinates is $-4-(-7) = 3$. The scale factor $k$ can be found by comparing the distances of corresponding points from the center of dilation. If we consider the ratio of the distance of the image point from the center of dilation to the distance of the original point from the center of dilation. The scale factor $k=\frac{1}{3}$ because if we assume the center of dilation is at $x=- 5$ (mid - way between some corresponding points on the line $y = 3$), the distance from $K(-7,3)$ to $(-5,3)$ is $2$ and the distance from $K'(-4,3)$ to $(-5,3)$ is $\frac{2}{3}$.

Step3: Find the center of dilation

We can use the formula for dilation. For a dilation with scale factor $k$ and center $(a,b)$ and a point $(x,y)$ and its image $(x',y')$, the formula is $x'=a + k(x - a)$ and $y'=b + k(y - b)$. Since the $y$ - coordinates of corresponding points are the same ($y=y' = 3$ for $L$ and $L'$ and $K$ and $K'$), the center of dilation lies on the line $y = 3$. Let's use the $x$ - coordinates of $L(-2,3)$ and $L'(1,3)$. Let the $x$ - coordinate of the center be $a$. Then $1=a+\frac{1}{3}(-2 - a)$. Solving for $a$:
\[

$$\begin{align*} 1&=a-\frac{2}{3}-\frac{1}{3}a\\ 1+\frac{2}{3}&=\frac{2}{3}a\\ \frac{5}{3}&=\frac{2}{3}a\\ a&=- 5 \end{align*}$$

\]
So the center of dilation is $(-5,3)$.

Answer:

scale factor: $\frac{1}{3}$
center of the dilation: $(-5,3)$