https://file.aiquickdraw.com/mathbot/file/1724910187446dbdq2ptl.jpg

determine a medida do perímetro do triângulo, sabendo que DE é paralelo a ВС.

To determine the perimeter of the triangle \( ABC \) in the image, let's analyze the given information and apply some geometric properties.

### Step 1: Understanding the Triangle and Given Dimensions
- \( AB = 6 \) cm
- \( BC = 8 \) cm
- \( CA = 9 \) cm
- \( DE \) is parallel to \( BC \), and the length \( AE = 7.5 \) cm.

### Step 2: Apply the Concept of Similarity
Since \( DE \) is parallel to \( BC \), triangles \( ADE \) and \( ABC \) are similar by the Basic Proportionality Theorem (Thales' theorem). This implies that the sides of these triangles are proportional:
\[
\frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC}
\]
Let's denote \( AD = x \), \( DE = y \), and \( BC = 8 \) cm. Since \( AE = 7.5 \) cm and \( AC = 9 \) cm, the ratio of the sides can be written as:
\[
\frac{AE}{AC} = \frac{7.5}{9} = \frac{5}{6}
\]
Therefore:
\[
\frac{AD}{AB} = \frac{5}{6}
\]
\[
AD = \frac{5}{6} \times 6 = 5 \text{ cm}
\]

### Step 3: Calculate the Perimeter of Triangle \( ABC \)
The perimeter of triangle \( ABC \) is simply the sum of its sides:
\[
\text{Perimeter} = AB + BC + CA = 6 \text{ cm} + 8 \text{ cm} + 9 \text{ cm} = 23 \text{ cm}
\]

Thus, the perimeter of triangle \( ABC \) is **23 cm**.

Would you like more details, or do you have any other questions?

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### Additional Questions for Exploration:
1. How would you calculate the area of triangle \( ABC \) using Heron's formula?
2. If \( DE \) is half the length of \( BC \), what would be the ratio of the areas of triangles \( ADE \) and \( ABC \)?
3. How does the Basic Proportionality Theorem (Thales' theorem) apply in different geometric scenarios?
4. What is the relationship between similar triangles and their corresponding angles?
5. Can the triangle's circumcenter be found using the given sides? If so, how?

### Tip:
When dealing with similar triangles, remember that their corresponding angles are equal, and their corresponding sides are in proportion. This property is essential for solving problems involving parallel lines and proportional segments.

Learn how to calculate the perimeter of triangle ABC with DE parallel to BC. Understand the application of Thales' theorem and the concept of similarity in geometry. Suitable for Grades 9-12 students studying geometric properties of triangles.

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