Select the correct answer.
A pair of trapezoids ABCD (Bottom) PQRS (Top) are shown. The coordinates of preimage are (2,2) , (6,6) , (12,8), and (16,2). The coordinates of the image are (11,15), (9,12), (6,12) , and (4,15).&nbsp;&nbsp;

Polygon ABCD is reflected and dilated to give polygon PQRS. The coordinates of the preimage are (2, 2), (6, 8), (12, 8), and (16, 2). The coordinates of the image are (11, 15), (9, 12), (6, 12), and (4, 15). What is the scale factor of the dilation?

A.
0.25

B.
0.33

C.
0.5

D.
0.75

To solve for the scale factor of dilation, we need to calculate the ratio of the distances between corresponding points on the preimage and the image. Dilation changes the size of a figure by multiplying the distances between points by a constant scale factor.

### Step 1: Calculate the distance between two corresponding points in the preimage and image.

Let's take two corresponding points:
- Preimage point A (2, 2) and image point P (11, 15).

We can calculate the distance between these two points using the distance formula:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
For the preimage point \( (2, 2) \) and the image point \( (11, 15) \):
\[
\text{Distance} = \sqrt{(11 - 2)^2 + (15 - 2)^2} = \sqrt{9^2 + 13^2} = \sqrt{81 + 169} = \sqrt{250} \approx 15.81
\]

### Step 2: Now calculate the distance between another pair of corresponding points.

Preimage point B (6, 8) and image point Q (9, 12):
\[
\text{Distance} = \sqrt{(9 - 6)^2 + (12 - 8)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]

### Step 3: Find the ratio of these distances.

The scale factor is the ratio of the image distance to the preimage distance. Using the points from Step 2:
\[
\text{Scale factor} = \frac{5}{10} = 0.5
\]

### Conclusion:
The scale factor of the dilation is \( 0.5 \).

So, the correct answer is:
**C. 0.5**

Would you like more details on any part of this solution?

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**5 Related Questions:**
1. How do you find the center of dilation in a transformation?
2. What is the relationship between dilation and similarity of shapes?
3. Can a dilation have a negative scale factor? If so, what does it represent?
4. How would the coordinates of the image change if the scale factor was 0.75?
5. How do reflections affect the orientation of a figure in the coordinate plane?

**Tip:** When calculating distances, make sure to double-check your coordinate points, especially when they involve reflection or translation before dilation.

Polygon ABCD is reflected and dilated to give polygon PQRS. The coordinates of the preimage are (2, 2), (6, 8), (12, 8), and (16, 2). The coordinates of the image are (11, 15), (9, 12), (6, 12), and (4, 15). What is the scale factor of the dilation?

A. 0.25
B. 0.33
C. 0.5
D. 0.75

Calculate the scale factor of dilation for trapezoids ABCD and PQRS. The preimage coordinates are (2, 2), (6, 8), (12, 8), and (16, 2), while the image coordinates are (11, 15), (9, 12), (6, 12), and (4, 15). Suitable for students in Grades 8-10, this problem focuses on coordinate geometry, reflection, and dilation.

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