https://file.aiquickdraw.com/mathbot/file/1728501466672y7o8zeni.jpg

The two shapes shown in the image are pentagons, and it appears that one pentagon is a scaled-up version of the other, meaning they are similar figures.

To find the length of \( ST \) in the larger pentagon, we can use the property of similar figures: corresponding sides of similar shapes are proportional. In this case, the two pentagons are proportional, so the ratio of corresponding side lengths in the smaller pentagon to those in the larger pentagon will be constant.

### Identifying Corresponding Sides:
- In the smaller pentagon, side \( AB = 11 \, \text{cm} \) corresponds to side \( RS = 33 \, \text{cm} \) in the larger pentagon.
- Side \( BC \) in the smaller pentagon corresponds to side \( ST \) in the larger pentagon, and we are asked to find the length of \( ST \).

### Finding the Scale Factor:
The ratio of the corresponding sides \( AB \) and \( RS \) gives the scale factor between the smaller and larger pentagons:
\[
\text{Scale factor} = \frac{RS}{AB} = \frac{33 \, \text{cm}}{11 \, \text{cm}} = 3
\]

### Calculating the Length of \( ST \):
The corresponding side in the smaller pentagon, \( BC \), is 4.4 cm. To find the length of \( ST \) in the larger pentagon, we multiply the length of \( BC \) by the scale factor:
\[
ST = BC \times 3 = 4.4 \, \text{cm} \times 3 = 13.2 \, \text{cm}
\]

### Final Answer:
The length of \( ST \) is \( 13.2 \, \text{cm} \).

Would you like more details or have any questions?

Here are some related questions:
1. What is the perimeter of the larger pentagon?
2. How would you calculate the area of these similar pentagons?
3. Can the ratio of the areas of these pentagons be found using the scale factor?
4. How can we check the angles in similar shapes to ensure similarity?
5. How can similar triangles or polygons be used in real-world applications?

**Tip:** When working with similar shapes, always look for corresponding sides or angles and apply the proportionality property carefully.

Solve the problem of finding the length of ST in a pair of similar pentagons using the scale factor and proportionality of similar figures. This problem involves applying geometry concepts and proportional reasoning, suitable for students in Grades 7-9.

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