One of the internal angles of the rhombus is 1200 and the short- er diagonal is 3.4 meters long. Find the perimeter of the rhombus

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.

Discover how to calculate the perimeter of a rhombus with an internal angle of 120° and a shorter diagonal of 3.4 meters. Learn about the properties of rhombuses, the law of cosines, and trigonometric calculations. Suitable for high school students studying geometry and trigonometry.

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