To solve the problem, we perform the dilation of the polygon ABCD using the given scale factor of $\frac{1}{2}$, with the origin as the center of dilation. Each vertex's coordinates will be scaled by multiplying them by $\frac{1}{2}$. Let's determine the new coordinates for each point:

Step 1: Extract the coordinates of the vertices

From the graph:

• $A(0, 0)$
• $B(2, 2)$
• $C(4, 0)$
• $D(2, -2)$

Step 2: Apply the dilation

Using the formula for dilation at the origin: $(x', y') = \left( \frac{1}{2}x, \frac{1}{2}y \right)$

1. $A(0, 0) \to A'(0, 0)$
2. $B(2, 2) \to B'(1, 1)$
3. $C(4, 0) \to C'(2, 0)$
4. $D(2, -2) \to D'(1, -1)$

Step 3: Resulting coordinates of AʹBʹCʹDʹ

The coordinates of the dilated polygon are: $A'(0, 0), B'(1, 1), C'(2, 0), D'(1, -1)$

Would you like further explanations or clarifications on how dilation works?

Here are 5 related questions to deepen your understanding:

1. How would the coordinates change if the scale factor was $2$ instead of $\frac{1}{2}$?
2. What happens to the shape of a polygon under dilation?
3. How do you determine the center of dilation from given coordinates?
4. Can dilation result in a reflection? If yes, under what conditions?
5. How is dilation different from translation or rotation?

Tip: Always check the proportionality of distances from the center of dilation to ensure the transformation is correct!