Question

k = 3
k = 1/4
k = 2
k = 1/2
k = 4
k = 1/3

Explanation:

Step1: Recall dilation formula

If a point $(x,y)$ is dilated by a scale - factor $k$ with the origin as the center of dilation, the new point $(x',y')$ is given by $(x',y')=(kx,ky)$. Let's consider two corresponding points, for example, point $D(-2, - 4)$ and its image $D'(-1,-2)$.

Step2: Calculate the scale - factor

We use the formula for dilation. For the $x$ - coordinates: $x' = kx$. Substitute $x=-2$ and $x'=-1$ into the formula $x' = kx$. Then $-1=k\times(-2)$. Solving for $k$, we get $k=\frac{-1}{-2}=\frac{1}{2}$. We can also check with the $y$ - coordinates. For $y$ - coordinates: $y' = ky$. Substitute $y = - 4$ and $y'=-2$ into the formula $y' = ky$. Then $-2=k\times(-4)$, and $k=\frac{-2}{-4}=\frac{1}{2}$.

Answer:

$k = \frac{1}{2}$