To perform a dilation of scale factor $\frac{1}{2}$ about the origin, we need to adjust each vertex of the figure by multiplying its coordinates by $\frac{1}{2}$.

Here are the steps:

Step 1: Identify Coordinates of Vertices

From the image:

• $I = (4, -2)$
• $J = (4, 2)$
• $K = (2, 2)$
• $L = (2, -2)$

Step 2: Apply Dilation Formula

The dilation formula is: $(x', y') = \left( \frac{1}{2} \cdot x, \frac{1}{2} \cdot y \right)$

Now, let's apply this to each vertex:

1. For point $I(4, -2)$: $I' = \left( \frac{1}{2} \cdot 4, \frac{1}{2} \cdot (-2) \right) = (2, -1)$
2. For point $J(4, 2)$: $J' = \left( \frac{1}{2} \cdot 4, \frac{1}{2} \cdot 2 \right) = (2, 1)$
3. For point $K(2, 2)$: $K' = \left( \frac{1}{2} \cdot 2, \frac{1}{2} \cdot 2 \right) = (1, 1)$
4. For point $L(2, -2)$: $L' = \left( \frac{1}{2} \cdot 2, \frac{1}{2} \cdot (-2) \right) = (1, -1)$

Step 3: Plot the New Coordinates

The new vertices after dilation are:

• $I' = (2, -1)$
• $J' = (2, 1)$
• $K' = (1, 1)$
• $L' = (1, -1)$

These points form the new figure, which is a smaller version of the original shape, reduced by a factor of $\frac{1}{2}$.

Would you like me to generate an image of the dilated shape or provide further details on this transformation?

Here are 5 related questions to deepen your understanding:

1. How do you determine the center of dilation?
2. What happens to the orientation of a shape during a dilation?
3. How would the result change if the scale factor was greater than 1?
4. What are the properties of dilations regarding angles and lengths?
5. Can you think of a real-world application where dilation transformations are useful?

Tip: In dilations, the shape preserves its orientation, and angles remain the same, but distances from the origin change based on the scale factor.