In a rhombus ABCD, `angle A = 60^(@)`, AB = 12cm. Then the diagonal BD is

Options

A. 10 cm
B. `2sqrt(3) cm`
C. 6 cm
D. 12 cm

Answer

To find the length of diagonal BD in rhombus ABCD, where angle A = 60° and side AB = 12 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the properties of a rhombus**:
- In a rhombus, the diagonals bisect each other at right angles (90°).
- The opposite angles are equal.

2. **Identify the angles**:
- Given that angle A = 60°, angle B will also be 60° (since opposite angles are equal).
- Therefore, angle D = 60° and angle C = 120° (since the sum of angles in a quadrilateral is 360°).

3. **Determine the angles at point O (the intersection of the diagonals)**:
- Since diagonals bisect each other at right angles, angle AOB = 90°.
- We can find angle AOB by splitting angle A into two equal parts: angle BAD = angle AOB = 30°.

4. **Use triangle AOB**:
- In triangle AOB, we know:
- AB = 12 cm (hypotenuse),
- Angle AOB = 90°,
- Angle OAB = 30°.
- We need to find OB (half of diagonal BD).

5. **Apply trigonometric ratios**:
- Using the sine function:
\[ \sin(30°) = \frac{OB}{AB} \]
- We know that \(\sin(30°) = \frac{1}{2}\). Therefore:
\[ \frac{1}{2} = \frac{OB}{12} \]
- Solving for OB:
\[ OB = 12 \times \frac{1}{2} = 6 \text{ cm} \]

6. **Calculate diagonal BD**:
- Since O is the midpoint of diagonal BD, we have:
\[ BD = 2 \times OB = 2 \times 6 = 12 \text{ cm} \]

### Final Answer:
The length of diagonal BD is **12 cm**.

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