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Determine o valor de x  o perímetro do triângulo mede 75m

Let's reevaluate the problem with the given conditions and the diagram to determine the value of \( x \) in the triangle \( ABC \).

### Given:
1. \( P \) is the midpoint of \( \overline{AB} \).
2. \( \overline{PQ} \parallel \overline{BC} \).
3. The perimeter of triangle \( ABC \) is 75 meters.
4. \( AP = 15 \) m, \( PB = 12.5 \) m (since \( P \) is the midpoint, \( AB = 2 \times 12.5 = 25 \) m).
5. \( x = BC \) (the side we need to find).

### Step 1: Use the information given
- Since \( P \) is the midpoint of \( AB \) and \( \overline{PQ} \parallel \overline{BC} \), triangles \( APQ \) and \( ABC \) are similar with a similarity ratio of 1:2. This implies \( PQ = \frac{BC}{2} \).

### Step 2: Calculate the perimeter equation
The perimeter of triangle \( ABC \) is the sum of its sides:
\[
AB + AC + BC = 75 \text{ m}
\]
Substitute \( AB = 25 \) m and let \( AC = y \) and \( BC = x \):
\[
25 + y + x = 75
\]
\[
y + x = 50 \text{ m}
\]

### Step 3: Use triangle similarity
Since triangles \( APQ \) and \( ABC \) are similar and \( \overline{PQ} \parallel \overline{BC} \), the ratio of the sides \( \frac{PQ}{BC} = \frac{1}{2} \). From the diagram:
\[
AP = 15 \text{ m}, \quad PB = 12.5 \text{ m}, \quad PQ = \frac{x}{2}
\]

### Step 4: Solve for \( x \) using the perimeter equation
Given that \( AC + BC = 50 \) m:
\[
y + x = 50 \text{ m}
\]
Since \( y = PQ + QC \) and \( PQ = \frac{x}{2} \), we substitute \( y = 15 + \frac{x}{2} \) into the perimeter equation:
\[
15 + \frac{x}{2} + x = 50
\]
Combine like terms:
\[
\frac{x}{2} + x = 35
\]
Multiply by 2 to eliminate the fraction:
\[
x + 2x = 70
\]
\[
3x = 70
\]
\[
x = \frac{70}{3} \approx 23.33 \text{ m}
\]

However, this answer does not satisfy the initial conditions, suggesting a misinterpretation of the lengths or perimeter constraints.

### Correct Approach:
Re-examine the similarity and calculate \( AC \) correctly. Use the correct ratio for similar triangles:
\[
AC = y = 2 \times AP = 2 \times 15 = 30 \text{ m}
\]
Thus,
\[
x + 30 = 50 \quad \Rightarrow \quad x = 20 \text{ m}
\]

The value of \( x \), considering the correct proportionality and constraints, is 20 meters.

---

Would you like any further clarifications or additional questions on this topic? Here are some related questions to consider:

1. How do you determine similarity between triangles?
2. What role does the perimeter play in solving triangle problems?
3. Can you solve a problem where \( P \) is not the midpoint?
4. How does the parallel line condition affect the triangle's similarity?
5. What would happen if \( AP \) were not given?

**Tip:** Double-check all given conditions and derived equations when solving for unknowns in geometric problems to ensure all steps align with the problem's constraints.

Explore how to determine the value of x in a triangle with a perimeter of 75 meters. Understand the concepts of triangle similarity and perimeter equations used in solving this geometric problem. Suitable for high school students.

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