To perform a dilation of the quadrilateral ABCD using point A as the center of dilation and a scale factor of $\frac{1}{2}$, we need to follow these steps:

1. Identify the coordinates of each vertex in the original quadrilateral ABCD.

   • Point $A$ is already at the origin (0, 0).
   • Point $B$ has coordinates (4, 0).
   • Point $C$ has coordinates (4, 2).
   • Point $D$ has coordinates (2, 2).

2. Apply the dilation formula to each point using $A$ as the center:

   • If the center of dilation is $A = (x_A, y_A)$ and the scale factor is $k$, then the new coordinates $(x', y')$ of any point $(x, y)$ are given by: $x' = x_A + k \times (x - x_A)$ $y' = y_A + k \times (y - y_A)$

   Since $A = (0, 0)$ and the scale factor $k = \frac{1}{2}$, the formula simplifies to: $x' = \frac{1}{2} \times x$ $y' = \frac{1}{2} \times y$

3. Calculate the new vertices:

   • For Point B (4, 0): $x'_B = \frac{1}{2} \times 4 = 2, \quad y'_B = \frac{1}{2} \times 0 = 0$ New Point $B' = (2, 0)$

   • For Point C (4, 2): $x'_C = \frac{1}{2} \times 4 = 2, \quad y'_C = \frac{1}{2} \times 2 = 1$ New Point $C' = (2, 1)$

   • For Point D (2, 2): $x'_D = \frac{1}{2} \times 2 = 1, \quad y'_D = \frac{1}{2} \times 2 = 1$ New Point $D' = (1, 1)$

4. Summarize the new coordinates of the vertices after the dilation:

   • $A' = (0, 0)$ (unchanged)
   • $B' = (2, 0)$
   • $C' = (2, 1)$
   • $D' = (1, 1)$

These are the coordinates of the vertices of the dilated quadrilateral $A'B'C'D'$.

Would you like to explore more details or have any questions? Here are some related questions you might find interesting:

1. What would happen if the scale factor were doubled to $2$ instead of $\frac{1}{2}$?
2. How would you describe the effect of dilation on the shape and area of the quadrilateral?
3. Can you determine the area of the original quadrilateral ABCD and the dilated quadrilateral A'B'C'D'?
4. What is the general effect of a dilation with a negative scale factor?
5. How do the side lengths of the quadrilateral change under dilation with a scale factor of $\frac{1}{2}$?

Tip: Always check if the center of dilation lies on any of the vertices. If it does, that vertex will not change its position after dilation.