https://file.aiquickdraw.com/mathbot/file/1725527493818sh3930zg.jpg

Let's analyze the image.

It shows two polygons, \( QRST \) and \( ABCD \), which are labeled as similar. The side lengths of these polygons are given:

- In polygon \( QRST \):
  - \( QR = 12 \)
  - \( RS = 28 \)
  - \( ST = 36 \)
  - \( TQ = 18 \)

- In polygon \( ABCD \):
  - \( AB = 6 \)
  - \( BC = 14 \)
  - \( CD = 18 \)
  - \( DA = 9 \)

The task asks to:
1. Complete the similarity statement: \( QRST \sim ABCD \).
2. Find the ratio of a side length in \( QRST \) to its corresponding side length in \( ABCD \).
3. Simplify the answer as a proper fraction, improper fraction, or whole number.

### Step-by-step solution:

1. **Similarity Statement**: The polygons are similar, so the corresponding sides are proportional. By examining the vertices, we can match the corresponding sides:
   - \( QR \leftrightarrow AB \)
   - \( RS \leftrightarrow BC \)
   - \( ST \leftrightarrow CD \)
   - \( TQ \leftrightarrow DA \)

   The complete similarity statement is:
   \[
   QRST \sim ABCD
   \]

2. **Ratio of Side Lengths**: We can calculate the ratio of corresponding sides by picking any corresponding side lengths. Let's use \( QR \) and \( AB \):
   \[
   \text{Ratio} = \frac{QR}{AB} = \frac{12}{6} = 2
   \]

Thus, the ratio of side lengths between the two polygons is \( 2 \).

Would you like further clarification on this solution?

Here are 5 related questions to consider:
1. What other ratios can we form using the corresponding sides of these polygons?
2. How would you find the ratio of the perimeters of the polygons?
3. What happens to the areas of two similar polygons when their side lengths have a ratio of 2?
4. Can you apply the same method to compare the diagonals of these polygons?
5. How does the concept of similarity apply in three-dimensional shapes?

**Tip:** In similar polygons, the ratio of corresponding sides is constant, and this ratio can also be used to compare other aspects like perimeter and area.

The two polygons below are similar. Complete the similarity statement QRST ~ ABCD. Find the ratio of a side length in the first polygon to its corresponding side length in the second polygon. Simplify your answer and write it as a proper fraction, improper fraction, or whole number.

Solve the problem of finding the ratio between the sides of two similar polygons QRST and ABCD. Learn how to calculate side length ratios and simplify them. This geometry problem is suitable for students in Grades 7-9.

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