If an equilateral triangle is inscribed in the circle `x^2 + y2 = a^2`, the length of its each side is

Options

A. `sqrt(2)a`
B. `(sqrt(3))/(2)a`
C. `sqrt(3)a`
D. none of these

Explanation

To find the length of each side of an equilateral triangle inscribed in the circle given by the equation \( x^2 + y^2 = a^2 \), we can follow these steps:

### Step 1: Understand the Circle and Triangle
The equation \( x^2 + y^2 = a^2 \) represents a circle with a radius \( a \). An equilateral triangle inscribed in this circle will have its vertices on the circumference of the circle.

**Hint:** Remember that the radius of the circle is the distance from the center to any point on the circle.

### Step 2: Identify the Angles
In an equilateral triangle, each angle measures \( 60^\circ \). When we draw lines from the center of the circle to the vertices of the triangle, these lines will bisect the angles of the triangle.

**Hint:** The angles formed at the center of the circle by the lines to the vertices will be \( 60^\circ \) each, leading to \( 30^\circ \) angles at the center.

### Step 3: Calculate the Length from Center to Midpoint
Let’s denote the vertices of the triangle as \( A, B, C \) and the center of the circle as \( O \). The line segments \( OA, OB, \) and \( OC \) are all equal to the radius \( a \). The midpoint of side \( BC \) can be denoted as \( M \).

The angle \( AOB \) at the center is \( 60^\circ \), so each angle \( AOM \) and \( BOM \) is \( 30^\circ \).

**Hint:** Use trigonometric functions to find the lengths of segments.

### Step 4: Use Trigonometry to Find PM
The length \( OM \) can be calculated using the cosine of \( 30^\circ \):
\[ OM = OA \cdot \cos(30^\circ) = a \cdot \frac{\sqrt{3}}{2} \]

**Hint:** Remember that \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \).

### Step 5: Relate PM to PQ
Since \( PM \) is half of the side \( PQ \) of the triangle, we can express \( PQ \) as:
\[ PQ = 2 \cdot OM = 2 \cdot \left(a \cdot \frac{\sqrt{3}}{2}\right) = a\sqrt{3} \]

**Hint:** The side length of the triangle can be found by doubling the length from the center to the midpoint of a side.

### Conclusion
Thus, the length of each side of the equilateral triangle inscribed in the circle is:
\[ \text{Length of each side} = a\sqrt{3} \]

### Final Answer
The correct option is \( \sqrt{3} \cdot a \).