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To match the dilation rules to the descriptions given in the table:

### Problem Analysis
1. **Triangle Dilation**: The description states that a triangle is dilated by a scale factor of \(\frac{1}{2}\). A dilation by a scale factor of \(\frac{1}{2}\) reduces each coordinate by half. Thus, the transformation rule should be:  
\[
(x, y) \rightarrow \left(\frac{1}{2}x, \frac{1}{2}y\right)
\]

2. **Square Dilation**: The description states that a square is dilated by a scale factor of 2. A dilation by a scale factor of 2 doubles each coordinate value. Thus, the transformation rule should be:  
\[
(x, y) \rightarrow (2x, 2y)
\]

### Filling in the Table
- **For the triangle**: \((x, y) \rightarrow \left(\frac{1}{2}x, \frac{1}{2}y\right)\)
- **For the square**: \((x, y) \rightarrow (2x, 2y)\)

### Final Answer
- **Triangle**: \((x, y) \rightarrow \left(\frac{1}{2}x, \frac{1}{2}y\right)\)
- **Square**: \((x, y) \rightarrow (2x, 2y)\)

Let me know if you'd like further details or have any questions. 

### Related Questions
1. What are other transformations besides dilation that can change a shape’s size and position?
2. How do you determine the center of dilation?
3. What effect does a negative scale factor have on a shape?
4. How does dilation differ from translation and reflection?
5. Can dilation change the shape’s orientation?

### Tip
When performing dilations, always check if the center of dilation is the origin, as this simplifies the transformation rule considerably.

The table describes two dilations with the center of dilation at the origin. Which rule describes each dilation?

This problem involves identifying the correct dilation rules for a triangle and square with respect to the origin. Learn about transformations, dilations, and their effects on shapes with detailed explanations for scale factors of 1/2 and 2.

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