https://file.aiquickdraw.com/mathbot/file/17263529931862g00r5w9.jpg

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.

Learn how to perform a dilation of a figure by a scale factor of 1/2 about the origin. This problem involves identifying the coordinates of the figure's vertices, applying the dilation formula, and plotting the new points to get the transformed figure. Suitable for students in Grades 8-10.

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