One of the four angles of a rhombus is `60^(@)`. If the length of each side of the rhombus is 8 cm, then the length of the longer diagonal is

Options

A. `8sqrt(3)cm`
B. 8 cm
C. `4sqrt(3) cm`
D. `(8)/(sqrt(3)) cm`

Answer

To find the length of the longer diagonal of a rhombus where one angle is \(60^\circ\) and each side measures 8 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Properties of a Rhombus**:
A rhombus has diagonals that bisect each other at right angles. The angles of the rhombus can be divided into two equal parts by the diagonals.

2. **Identifying the Angles**:
Given that one angle of the rhombus is \(60^\circ\), the opposite angle is also \(60^\circ\). The other two angles will be \(120^\circ\) each (since the sum of angles in a quadrilateral is \(360^\circ\)).

3. **Setting Up the Right Triangle**:
When we draw the diagonals, they will intersect at point \(O\), creating four right triangles. We will focus on one of the triangles formed by the angle \(30^\circ\) (half of \(60^\circ\)).

4. **Using the Sine Function**:
In triangle \(AOB\) (where \(A\) and \(B\) are the endpoints of one diagonal), we can use the sine function:
\[ \sin(30^\circ) = \frac{Opposite}{Hypotenuse} \]
Here, the opposite side is \(OC\) (half of one diagonal), and the hypotenuse is \(OA\) (which is the side of the rhombus, \(8 \, \text{cm}\)):
\[ \sin(30^\circ) = \frac{OC}{OA} \implies \frac{1}{2} = \frac{OC}{8} \]
Therefore, \(OC = 4 \, \text{cm}\).

5. **Finding the Length of One Diagonal**:
Since \(OC\) is half of one diagonal \(AC\), the full length of diagonal \(AC\) is:
\[ AC = 2 \times OC = 2 \times 4 = 8 \, \text{cm} \]

6. **Using the Cosine Function**:
Now we will use the cosine function for the triangle to find the other half of the other diagonal \(BD\):
\[ \cos(30^\circ) = \frac{Adjacent}{Hypotenuse} \]
Here, the adjacent side is \(OB\) (half of the other diagonal), and the hypotenuse is again \(OA\):
\[ \cos(30^\circ) = \frac{OB}{8} \implies \frac{\sqrt{3}}{2} = \frac{OB}{8} \]
Therefore, \(OB = 4\sqrt{3} \, \text{cm}\).

7. **Finding the Length of the Other Diagonal**:
Since \(OB\) is half of diagonal \(BD\), the full length of diagonal \(BD\) is:
\[ BD = 2 \times OB = 2 \times 4\sqrt{3} = 8\sqrt{3} \, \text{cm} \]

8. **Conclusion**:
The length of the longer diagonal \(BD\) is \(8\sqrt{3} \, \text{cm}\).

### Final Answer:
The length of the longer diagonal is \(8\sqrt{3} \, \text{cm}\).

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