To find the perimeter of the rhombus, we need to use the properties of the rhombus and the information given:

1. Internal Angle: One of the internal angles is $120^\circ$.
2. Shorter Diagonal: The shorter diagonal $d_1$ is $3.4$ meters.

Step 1: Use the diagonal properties of the rhombus

The diagonals of a rhombus bisect each other at right angles (90°). Let's denote:

• $d_1$ as the shorter diagonal $3.4$ meters.
• $d_2$ as the longer diagonal (which we need to find).
• $s$ as the side length of the rhombus.

Step 2: Relate the angle to the sides

The given internal angle is $120^\circ$. Since the diagonals bisect the angles of the rhombus, the angle between the diagonals will be half of $120^\circ$, which is $60^\circ$.

Step 3: Apply trigonometry to find the side length

Using the law of cosines in one of the triangles formed by the diagonals:

$s^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 - \left(\frac{d_1}{2}\right)\left(\frac{d_2}{2}\right)\cos(120^\circ)$

However, knowing that $\cos(120^\circ) = -\frac{1}{2}$, we simplify this further:

$s^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 + \left(\frac{d_1 \cdot d_2}{4}\right)$

Given $d_1 = 3.4$ meters, we can simplify further to find $d_2$. First, let’s determine $s$:

$\sin\left(\frac{120^\circ}{2}\right) = \sin(60^\circ) = \frac{\sqrt{3}}{2}$

Thus,

$s = \frac{d_1}{2 \times \sin\left(\frac{120^\circ}{2}\right)} = \frac{3.4}{2 \times \frac{\sqrt{3}}{2}} = \frac{3.4}{\sqrt{3}} \approx 1.962\ \text{meters}$

Step 4: Calculate the perimeter

The perimeter of a rhombus is $4$ times the side length:

$\text{Perimeter} = 4 \times s \approx 4 \times 1.962 = 7.848\ \text{meters}$

Final Answer

The perimeter of the rhombus is approximately 7.848 meters.

Do you want further details or have any questions?

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Here are 5 relative questions for further exploration:

1. How does changing the internal angle affect the perimeter of the rhombus?
2. What is the relationship between the diagonals of a rhombus and its area?
3. How would the perimeter change if the shorter diagonal were doubled?
4. Can you determine the length of the longer diagonal $d_2$?
5. What is the formula for finding the area of a rhombus using its diagonals?

Tip: In any rhombus, all sides are equal in length, but the angles can vary. This property can be useful in solving different types of problems involving rhombuses.