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The two images you uploaded relate to dilations in geometry. Let's work through the problems together.

### Explanation of Dilations:
A dilation is a transformation that enlarges or reduces a figure by a scale factor, with respect to a fixed point known as the center of dilation, which in these cases is the origin \((0, 0)\).

The formula for dilation is:
\[
(x', y') = (k \cdot x, k \cdot y)
\]
Where:
- \( (x', y') \) are the coordinates after dilation,
- \( k \) is the scale factor,
- \( (x, y) \) are the original coordinates before dilation.

Now, let's go through the specific exercises from the worksheets.

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### First Worksheet: Graph the pre-image and the image of the figure

#### 1. Dilation of 2 about the origin
Points: 
- \( V(-1, -2) \)
- \( W(0, 2) \)
- \( X(1, 2) \)
- \( Y(2, 1) \)

For a dilation of 2, multiply both \( x \) and \( y \) coordinates of each point by 2:
- \( V'(-2, -4) \)
- \( W'(0, 4) \)
- \( X'(2, 4) \)
- \( Y'(4, 2) \)

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#### 2. Dilation of 5 about the origin
Points:
- \( G(-1, 0) \)
- \( H(1, 1) \)
- \( I(1, 0) \)

For a dilation of 5:
- \( G'(-5, 0) \)
- \( H'(5, 5) \)
- \( I'(5, 0) \)

---

#### 3. Dilation of 0.5 about the origin
Points:
- \( B(-1, 1) \)
- \( C(2, 2) \)
- \( D(1, -1) \)

For a dilation of 0.5:
- \( B'(-0.5, 0.5) \)
- \( C'(1, 1) \)
- \( D'(0.5, -0.5) \)

---

#### 4. Dilation of 2 about the origin
Points:
- \( E(-2, -1) \)
- \( F(-2, 2) \)
- \( G(2, -1) \)

For a dilation of 2:
- \( E'(-4, -2) \)
- \( F'(-4, 4) \)
- \( G'(4, -2) \)

---

### Second Worksheet: Find the coordinates of the image after the transformation

#### 5. Dilation of \( \frac{1}{4} \) about the origin
Points:
- \( B(0, -1) \)
- \( C(0, 2) \)
- \( D(1, 1) \)

For a dilation of \( \frac{1}{4} \):
- \( B'(0, -0.25) \)
- \( C'(0, 0.5) \)
- \( D'(0.25, 0.25) \)

---

#### 6. Dilation of 5 about the origin
Points:
- \( C(0, -1) \)
- \( D(-1, 1) \)
- \( E(1, 0) \)

For a dilation of 5:
- \( C'(0, -5) \)
- \( D'(-5, 5) \)
- \( E'(5, 0) \)

---

#### 7. Dilation of 1.5 about the origin
Points:
- \( E(-2, 1) \)
- \( D(1, 3) \)
- \( C(2, -2) \)

For a dilation of 1.5:
- \( E'(-3, 1.5) \)
- \( D'(1.5, 4.5) \)
- \( C'(3, -3) \)

---

#### 8. Dilation of 2 about the origin
Points:
- \( H(0, 1) \)
- \( G(1, 1) \)
- \( F(1, -1) \)

For a dilation of 2:
- \( H'(0, 2) \)
- \( G'(2, 2) \)
- \( F'(2, -2) \)

---

Would you like further explanation on any of these problems? Here are 5 questions to explore the concept further:

1. How would the dilation change if the center was not the origin?
2. What happens when the scale factor is negative?
3. Can you explain the effect of a dilation with a scale factor between 0 and 1?
4. How do dilations affect the area of a figure?
5. What is the relationship between the original shape and its image after dilation?

### Tip:
Always multiply both the \( x \) and \( y \) coordinates by the scale factor to correctly calculate the image points under dilation.

Graph the pre-image and the image of the figure using the transformation given (dilation about the origin). Find the coordinates of the image of each figure after the given transformation.

Explore how to graph figures and their dilated images about the origin using specific scale factors. This exercise covers dilation formulas and their applications to transformations, suitable for students in Grades 9-10.

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