Math Problem Statement

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

Solution

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

Step 1: Understanding the Triangle and Given Dimensions

  • AB=6AB = 6 cm
  • BC=8BC = 8 cm
  • CA=9CA = 9 cm
  • DEDE is parallel to BCBC, and the length AE=7.5AE = 7.5 cm.

Step 2: Apply the Concept of Similarity

Since DEDE is parallel to BCBC, triangles ADEADE and ABCABC are similar by the Basic Proportionality Theorem (Thales' theorem). This implies that the sides of these triangles are proportional: ADAB=AEAC=DEBC\frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC} Let's denote AD=xAD = x, DE=yDE = y, and BC=8BC = 8 cm. Since AE=7.5AE = 7.5 cm and AC=9AC = 9 cm, the ratio of the sides can be written as: AEAC=7.59=56\frac{AE}{AC} = \frac{7.5}{9} = \frac{5}{6} Therefore: ADAB=56\frac{AD}{AB} = \frac{5}{6} AD=56×6=5 cmAD = \frac{5}{6} \times 6 = 5 \text{ cm}

Step 3: Calculate the Perimeter of Triangle ABCABC

The perimeter of triangle ABCABC is simply the sum of its sides: Perimeter=AB+BC+CA=6 cm+8 cm+9 cm=23 cm\text{Perimeter} = AB + BC + CA = 6 \text{ cm} + 8 \text{ cm} + 9 \text{ cm} = 23 \text{ cm}

Thus, the perimeter of triangle ABCABC is 23 cm.

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


Additional Questions for Exploration:

  1. How would you calculate the area of triangle ABCABC using Heron's formula?
  2. If DEDE is half the length of BCBC, what would be the ratio of the areas of triangles ADEADE and ABCABC?
  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.

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Math Problem Analysis

Mathematical Concepts

Geometry
Similarity of Triangles
Perimeter Calculation

Formulas

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Theorems

Basic Proportionality Theorem (Thales' theorem)

Suitable Grade Level

Grades 9-12