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

To determine the perimeter of the triangle given that \(DE\) is parallel to \(BC\), let's analyze the geometric properties and apply the information provided.

We are given the lengths of various sides in the triangle:

- \(AB = 6 \, \text{cm}\)
- \(AC = 8 \, \text{cm}\)
- \(BC = 9 \, \text{cm}\)
- \(AE = 7.5 \, \text{cm}\)
- \(EC = 9 \, \text{cm}\)

Given that \(DE \parallel BC\), the triangles \(ADE\) and \(ABC\) are similar by the Basic Proportionality Theorem (or Thales' Theorem).

To find the perimeter of the triangle \(ADE\), we need to calculate the lengths of \(DE\), \(AD\), and \(AE\).

Let's start by setting up the ratio of similarity:

\[
\frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC}
\]

Given the lengths of \(AE = 7.5 \, \text{cm}\) and \(AC = 8 \, \text{cm}\), the ratio is:

\[
\frac{AE}{AC} = \frac{7.5}{15} = \frac{1}{2}
\]



Learn how to determine the perimeter of a triangle when given that DE is parallel to BC. This geometry problem involves applying the Basic Proportionality Theorem (Thales' Theorem) and similarity concepts. Detailed solution provided with specific lengths and ratios used in the calculations.

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