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Let's go over the problems in your worksheet step by step. These problems involve applying a dilation transformation, where the figure is enlarged or reduced by a specific scale factor relative to the origin.

### 1. Dilation of 1.5 about the origin
You will multiply the coordinates of each vertex by the dilation factor 1.5.

For example, if a vertex is at \((x, y)\), the new coordinates after dilation will be \((1.5x, 1.5y)\).

### 2. Dilation of \( \frac{5}{2} \) about the origin
Multiply each vertex’s coordinates by \( \frac{5}{2} \) or 2.5.

### 3. Dilation of \( \frac{1}{2} \) about the origin
Multiply each vertex’s coordinates by \( \frac{1}{2} \) or 0.5.

### 4. Dilation of 0.25 about the origin
Multiply each vertex’s coordinates by 0.25.

### 5. Dilation of 2 about the origin
Multiply each vertex’s coordinates by 2.

### 6. Dilation of 2.5 about the origin
Multiply each vertex’s coordinates by 2.5.

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For the transformations below, let's find the new coordinates of the vertices.

### 11. Dilation of 2.5 about the origin
Vertices: I(-2, -1), J(-2, 2), K(2, 1)

- \( I' = 2.5 \times (-2, -1) = (-5, -2.5) \)
- \( J' = 2.5 \times (-2, 2) = (-5, 5) \)
- \( K' = 2.5 \times (2, 1) = (5, 2.5) \)

### 12. Dilation of 2 about the origin
Vertices: T(-1, -1), U(0, 1), V(1, 0), W(1, -1)

- \( T' = 2 \times (-1, -1) = (-2, -2) \)
- \( U' = 2 \times (0, 1) = (0, 2) \)
- \( V' = 2 \times (1, 0) = (2, 0) \)
- \( W' = 2 \times (1, -1) = (2, -2) \)

### 13. Dilation of \( \frac{1}{4} \) about the origin
Vertices: Y(-2, 0), X(2, 2), W(1, -2)

- \( Y' = \frac{1}{4} \times (-2, 0) = (-0.5, 0) \)
- \( X' = \frac{1}{4} \times (2, 2) = (0.5, 0.5) \)
- \( W' = \frac{1}{4} \times (1, -2) = (0.25, -0.5) \)

### 14. Dilation of 5 about the origin
Vertices: Y(1, -1), X(0, 1), W(1, 1)

- \( Y' = 5 \times (1, -1) = (5, -5) \)
- \( X' = 5 \times (0, 1) = (0, 5) \)
- \( W' = 5 \times (1, 1) = (5, 5) \)

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Would you like more details on any specific question? Here are some relative questions:

1. How do you determine the scale factor for dilation transformations?
2. How do you handle negative coordinates in dilations?
3. What is the general effect of a dilation with a factor greater than 1 versus less than 1?
4. How can you graphically represent dilation transformations?
5. What are the differences between dilation and other transformations like translation or rotation?

**Tip**: Always double-check your new coordinates after a dilation by comparing the distance from the origin; it should increase or decrease according to the scale factor!

Learn how to apply dilation transformations to figures in coordinate geometry with various scale factors. This guide provides a detailed step-by-step explanation of how to calculate the new coordinates after dilations, suitable for students in Grades 7-9.

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