If a triangle is inscribed in a circle of radius r , then which of the following triangle can have maximum area

Options

A. equilateral triangle with side `sqrt3r`
B. equilateral triangle with side `sqrt2r`
C. isosceles triangle with side `2r`
D. right angle triangle with side `r , 2r`

Answer

To determine which triangle inscribed in a circle of radius \( r \) has the maximum area, we can analyze the properties of different types of triangles.

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to find which triangle inscribed in a circle of radius \( r \) has the maximum area. The options provided include an equilateral triangle, a right triangle, and an isosceles triangle.

2. **Area of a Triangle Inscribed in a Circle**: The area \( A \) of a triangle inscribed in a circle can be expressed using the formula:
\[ A = \frac{abc}{4R} \]
where \( a, b, c \) are the lengths of the sides of the triangle, and \( R \) is the radius of the circumcircle.

3. **Equilateral Triangle**: For an equilateral triangle with side length \( a \):
- The circumradius \( R \) is given by:
\[ R = \frac{a}{\sqrt{3}} \]
- The area \( A \) of the equilateral triangle can be calculated as:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
- Substituting \( a = R \sqrt{3} \):
\[ A = \frac{\sqrt{3}}{4} (R \sqrt{3})^2 = \frac{3\sqrt{3}}{4} R^2 \]

4. **Right Triangle**: For a right triangle inscribed in a circle, the hypotenuse is the diameter of the circle. The area can be calculated but typically does not yield a larger area than the equilateral triangle for the same circumradius.

5. **Isosceles Triangle**: For an isosceles triangle, the area can be maximized, but it still does not exceed the area of the equilateral triangle inscribed in the same circle.

6. **Conclusion**: The equilateral triangle provides the maximum area when inscribed in a circle. Therefore, the triangle with maximum area is the equilateral triangle with side length \( \sqrt{3}R \).

### Final Answer:
The triangle that can have the maximum area inscribed in a circle of radius \( r \) is the **equilateral triangle** with side length \( \sqrt{3}R \).