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In this problem, we are given a triangle \(ABC\), where segment \(BD\) is both the altitude and the angle bisector of triangle \(ABC\). The sides of the triangle are represented as algebraic expressions in terms of \(x\).

- \(AB = 2x - 3\)
- \(BC = x + 5\)
- \(AC = 2 \cdot (x - 3)\)

We are tasked with calculating the perimeter of triangle \(ABC\) in units of length (u.c.).

### Step 1: Express the lengths of the sides
The perimeter \(P\) of a triangle is the sum of the lengths of all its sides:

\[
P = AB + BC + AC
\]

Now, let's substitute the expressions for each side:
\[
P = (2x - 3) + (x + 5) + 2(x - 3)
\]

### Step 2: Simplify the perimeter expression
Simplify the above expression by combining like terms:

\[
P = (2x - 3) + (x + 5) + (2x - 6)
\]

\[
P = 2x + x + 2x - 3 + 5 - 6
\]

\[
P = 5x - 4
\]

### Step 3: Conclusion
The perimeter of triangle \(ABC\) is:

\[
P = 5x - 4 \text{ units of length (u.c.)}.
\]

Would you like to explore this further or solve for \(x\) if more information becomes available? Here are some related questions to think about:

1. How does the property of being both an altitude and angle bisector affect the triangle's geometry?
2. What specific value of \(x\) would yield the perimeter if more conditions were provided?
3. How can you calculate the area of triangle \(ABC\) given that \(BD\) is the altitude?
4. What are the implications if \(BD\) is also a median?
5. How does the triangle change if \(BD\) divides the base \(AC\) in half?

**Tip:** In geometry, knowing that a segment serves as both an altitude and a bisector often points to specific triangle types, like isosceles triangles.

Sabendo que, do triângulo ABC, BD é a altura relativa ao lado AC e também é a bissetriz relativa ao ângulo ABC, calcule a medida do perímetro do triângulo ABC, em unidades de comprimento (u.c.).

Learn how to calculate the perimeter of triangle ABC where BD serves as both an altitude and an angle bisector. This solution involves key geometry concepts and algebraic simplification, suitable for students in Grades 9-12.

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