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.