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. `8 sqrt 3`cm
B. `8 cm`
C. `4sqrt3` cm
D. `8/sqrt3 cm`

Answer

To find the length of the longer diagonal of the rhombus, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Angles of the Rhombus**:
- 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 the angles in a rhombus is \(360^\circ\).

2. **Draw the Rhombus**:
- Label the vertices of the rhombus as \(A\), \(B\), \(C\), and \(D\) such that \(\angle ABC = 60^\circ\) and \(\angle BCD = 120^\circ\).

3. **Use the Properties of the Rhombus**:
- All sides of a rhombus are equal. Therefore, \(AB = BC = CD = DA = 8 \, \text{cm}\).

4. **Identify the Diagonals**:
- Let the diagonals be \(AC\) and \(BD\). The diagonal \(AC\) will be opposite the \(120^\circ\) angle, making it the longer diagonal.

5. **Apply the Cosine Rule in Triangle \(ADC\)**:
- In triangle \(ADC\), we can use the cosine rule:
\[ AC^2 = AD^2 + DC^2 - 2 \cdot AD \cdot DC \cdot \cos(\angle ADC) \]
- Here, \(AD = 8 \, \text{cm}\), \(DC = 8 \, \text{cm}\), and \(\angle ADC = 120^\circ\).

6. **Substitute the Values**:
- Using the cosine value:
\[ \cos(120^\circ) = -\frac{1}{2} \]
- Substitute into the formula:
\[ AC^2 = 8^2 + 8^2 - 2 \cdot 8 \cdot 8 \cdot \left(-\frac{1}{2}\right) \]
\[ AC^2 = 64 + 64 + 64 \]
\[ AC^2 = 192 \]

7. **Calculate the Length of the Diagonal**:
- Taking the square root:
\[ AC = \sqrt{192} = \sqrt{64 \cdot 3} = 8\sqrt{3} \, \text{cm} \]

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

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