Each of the two circles of same radius a passes through the centre of the other. If the circles cut each other at the points A, B and 0,0' be their centres, then area of the quadrilateral AOBO' is

Options

A. `1/4a^2`
B. `1/2 a^2`
C. `sqrt(3)/2a^2`
D. `a^2`

Answer

To find the area of the quadrilateral AOBO', we will follow these steps:

### Step 1: Understand the Configuration
We have two circles of the same radius \( a \) that pass through each other's centers. Let \( O \) and \( O' \) be the centers of the two circles. The circles intersect at points \( A \) and \( B \).

### Step 2: Identify the Shape
The quadrilateral \( AOBO' \) consists of the points \( A \), \( O \), \( B \), and \( O' \). Since both circles have the same radius and intersect at points \( A \) and \( B \), the segments \( OA \), \( OB \), \( O'A \), and \( O'B \) are all equal to the radius \( a \).

### Step 3: Determine the Triangles
The quadrilateral \( AOBO' \) can be divided into two triangles: \( AOB \) and \( AO'B \). Both of these triangles are equilateral triangles because all sides are equal to \( a \).

### Step 4: Calculate the Area of One Triangle
The area \( A \) of an equilateral triangle with side length \( s \) is given by the formula:
\[ A = \frac{\sqrt{3}}{4} s^2 \]
In our case, the side length \( s = a \). Therefore, the area of triangle \( AOB \) is:
\[ A_{AOB} = \frac{\sqrt{3}}{4} a^2 \]

### Step 5: Calculate the Area of the Other Triangle
Since triangle \( AO'B \) is also equilateral with the same side length \( a \), its area is the same:
\[ A_{AO'B} = \frac{\sqrt{3}}{4} a^2 \]

### Step 6: Calculate the Total Area of Quadrilateral AOBO'
The total area of quadrilateral \( AOBO' \) is the sum of the areas of triangles \( AOB \) and \( AO'B \):
\[ \text{Area}_{AOBO'} = A_{AOB} + A_{AO'B} = \frac{\sqrt{3}}{4} a^2 + \frac{\sqrt{3}}{4} a^2 = \frac{\sqrt{3}}{2} a^2 \]

### Final Answer
Thus, the area of the quadrilateral \( AOBO' \) is:
\[ \text{Area}_{AOBO'} = \frac{\sqrt{3}}{2} a^2 \]