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### Problems 1-3 (Quadrilateral ABCD dilation):
1. **What is the scale factor of the dilation?**
   The dilation moves point \( B(2,2) \) to point \( B'(4,4) \). To find the scale factor, use the ratio of the new coordinates to the original ones:
   \[
   \text{Scale Factor} = \frac{B'(4,4)}{B(2,2)} = 2
   \]
   So, the scale factor is **2**.

2. **Draw quadrilateral A'B'C'D'.**
   Since the scale factor is 2, each vertex of the quadrilateral ABCD will be multiplied by 2, resulting in new coordinates:
   - \( A(-2, 3) \to A'(-4, 6) \)
   - \( B(2, 2) \to B'(4, 4) \)
   - \( C(1, -3) \to C'(2, -6) \)
   - \( D(-2, -3) \to D'(-4, -6) \)

3. **Label the coordinate points for A', C', and D'.**
   From the dilation:
   - \( A'(-4, 6) \)
   - \( C'(2, -6) \)
   - \( D'(-4, -6) \)

### Problems 4-6 (Triangle dilation):
4. **Using the origin as the center and a scale factor of 2, draw the dilation of triangle ABC and label it DEF.**
   You’ll multiply each coordinate of triangle ABC by the scale factor 2:
   - \( A(5, 0) \to D(10, 0) \)
   - \( B(0, -5) \to E(0, -10) \)
   - \( C(-5, 0) \to F(-10, 0) \)

5. **Using the origin as the center and a scale factor of \( \frac{1}{2} \), draw the dilation of triangle ABC and label it GHI.**
   This time, you'll multiply the coordinates by \( \frac{1}{2} \):
   - \( A(5, 0) \to G(2.5, 0) \)
   - \( B(0, -5) \to H(0, -2.5) \)
   - \( C(-5, 0) \to I(-2.5, 0) \)

6. **Triangle GHI is a dilation of triangle DEF. Identify the center of dilation and the scale factor.**
   The center of dilation is still the origin, and the scale factor is the ratio of GHI to DEF. Since \( GHI \) was obtained by halving triangle ABC and DEF was obtained by doubling ABC, the scale factor is \( \frac{1}{4} \).

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Would you like more details on any step? Here are five related questions to explore further:

1. How do you calculate a dilation when the center of dilation is not the origin?
2. What happens to the area of a shape when it is dilated by a factor of \( k \)?
3. Can a dilation change the orientation of a shape?
4. How do you dilate a shape with respect to a different point than the origin?
5. What are the effects of negative scale factors in dilation?

### Tip:
Always verify your dilation by comparing the distances from the center of dilation. All distances should scale by the same factor.

Problems 1-6 involve dilating shapes using the origin as the center, finding scale factors, and labeling new coordinates.

This lesson covers how to perform dilations of quadrilaterals and triangles on a coordinate plane using the origin as the center. Learn how to find scale factors, label coordinates, and understand the concept of dilating shapes by factors such as 2 and 1/2.

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