A rhombus OABC is drawn inside a circle whose centre is at O in such a way that the vertices A, B and C of the rhombus are on the circle. If the area of the rhombus is `32sqrt(3) m^(2)`, then the radius of the circle is

Options

A. 64 m
B. 8 m
C. 3 m
D. 46 m

Answer

To find the radius of the circle in which the rhombus OABC is inscribed, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a rhombus OABC inscribed in a circle with center O. The vertices A, B, and C are on the circle. We know the area of the rhombus is \(32\sqrt{3} \, m^2\).

2. **Identifying the Properties of the Rhombus**:
In a rhombus, all sides are equal. Let the radius of the circle be \(r\). Therefore, the lengths of the sides OA, OB, OC (which are also the sides of the rhombus) are equal to \(r\).

3. **Using the Area of the Rhombus**:
The area \(A\) of a rhombus can also be expressed in terms of its diagonals \(d_1\) and \(d_2\):
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
However, we will also use the property of triangles formed by the diagonals.

4. **Dividing the Rhombus into Triangles**:
The diagonals of the rhombus divide it into four triangles of equal area. Thus, the area of the rhombus can be expressed as:
\[ \text{Area of Rhombus} = 2 \times \text{Area of Triangle OAB} \]
Given that the area of the rhombus is \(32\sqrt{3} \, m^2\), we have:
\[ 2 \times \text{Area of Triangle OAB} = 32\sqrt{3} \]
Therefore, the area of Triangle OAB is:
\[ \text{Area of Triangle OAB} = 16\sqrt{3} \, m^2 \]

5. **Calculating the Area of Triangle OAB**:
The area of triangle OAB can also be expressed using the formula:
\[ \text{Area} = \frac{1}{2} \times OA \times OB \times \sin(\angle AOB) \]
Since OA = OB = r, we can write:
\[ \text{Area of Triangle OAB} = \frac{1}{2} \times r \times r \times \sin(\angle AOB) = \frac{r^2}{2} \sin(\angle AOB) \]

6. **Setting Up the Equation**:
From the area of Triangle OAB:
\[ \frac{r^2}{2} \sin(\angle AOB) = 16\sqrt{3} \]
Thus, we can rearrange this to find:
\[ r^2 \sin(\angle AOB) = 32\sqrt{3} \]

7. **Finding the Value of r**:
Since the rhombus is symmetric and inscribed in a circle, the angle AOB can be calculated as \(60^\circ\) (as the rhombus divides the circle into equal angles). Therefore, \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\):
\[ r^2 \cdot \frac{\sqrt{3}}{2} = 32\sqrt{3} \]
Simplifying this gives:
\[ r^2 = 32\sqrt{3} \cdot \frac{2}{\sqrt{3}} = 64 \]
Thus, taking the square root:
\[ r = \sqrt{64} = 8 \, m \]

### Final Answer:
The radius of the circle is \(8 \, m\).