To find the perimeter of the triangle, we first need to ensure the side lengths satisfy the triangle inequality and that the areas calculated using the height corresponds to valid triangles.
The sides of the triangle are:
- \( a = 2x + 1 \)
- \( b = 4x - 3 \)
- \( c = 6x - 6 \)
We must also determine the relationship between the base and height of the triangle. The area \( A \) of the triangle can be computed as:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Given the height is 4 units, we can choose one of the sides as the base to calculate area. Let's take side \( c = 6x - 6 \) as the base.
Then the area can also be represented as:
\[ A = \frac{1}{2} \times (6x - 6) \times 4 = 2(6x - 6) = 12x - 12 \]
Now we will also find the area using Heron's formula. First, we should confirm the side lengths satisfy the triangle inequalities:
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\( (2x + 1) + (4x - 3) > (6x - 6) \) \[ 6x - 2 > 6x - 6 \quad\Rightarrow\quad -2 > -6 \quad\text{(True)} \]
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\( (2x + 1) + (6x - 6) > (4x - 3) \) \[ 8x - 5 > 4x - 3 \quad\Rightarrow\quad 4x > 2 \quad\Rightarrow\quad x > \frac{1}{2} \quad\text{(True)} \]
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\( (4x - 3) + (6x - 6) > (2x + 1) \) \[ 10x - 9 > 2x + 1 \quad\Rightarrow\quad 8x > 10 \quad\Rightarrow\quad x > \frac{5}{4} \quad\text{(True)} \]
The most restrictive condition is \( x > \frac{5}{4} \).
Next, calculate the perimeter:
\[ \text{Perimeter} = (2x + 1) + (4x - 3) + (6x - 6) \]
Combine the terms:
\[ = 2x + 1 + 4x - 3 + 6x - 6 = (2x + 4x + 6x) + (1 - 3 - 6) = 12x - 8 \]
Thus, the perimeter of the triangle is:
\[ \boxed{(12x - 8) \text{ units.}} \]
So the correct answer is \(B. (12x - 8) \units.\)
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