The triangle of maximum area that can be inscribed in a given circle of radius 'r' is

Options

A. An isoscles triangle with base equal to 2r
B. An equilateral of height `(2r)/3`
C. an equilateral trianlge having each of its side of length `sqrt3r`
D. `A right angle triangle having two of its sides of length 2r and r.

Answer

To determine the triangle of maximum area that can be inscribed in a given circle of radius \( r \), we can follow these steps:

### Step 1: Understand the Problem
We need to find out which triangle inscribed in a circle has the maximum area. The circle has a radius \( r \).

**Hint:** Recall that the area of a triangle can be influenced by the angles and the lengths of its sides.

### Step 2: Use the Formula for Area of a Triangle
The area \( A \) of a triangle inscribed in a circle can be expressed in terms of its sides \( a, b, c \) and the circumradius \( R \) (which is the radius of the circle). The formula is:
\[ A = \frac{abc}{4R} \]
For our case, since \( R = r \), we have:
\[ A = \frac{abc}{4r} \]

**Hint:** Consider the relationship between the angles of the triangle and the circumradius.

### Step 3: Analyze Different Types of Triangles
We can analyze various types of triangles to see which one gives the maximum area:
1. **Equilateral Triangle**
2. **Right Triangle**
3. **Isosceles Triangle**

**Hint:** Think about the properties of these triangles and how they relate to the angles subtended at the center of the circle.

### Step 4: Calculate the Area of an Equilateral Triangle
For an equilateral triangle inscribed in a circle of radius \( r \):
- The side length \( s \) of the triangle can be found using the formula:
\[ s = r \sqrt{3} \]
- The area \( A \) of the equilateral triangle is given by:
\[ A = \frac{\sqrt{3}}{4} s^2 = \frac{\sqrt{3}}{4} (r \sqrt{3})^2 = \frac{3\sqrt{3}}{4} r^2 \]

**Hint:** Verify if this area is indeed the maximum by comparing it with other triangle types.

### Step 5: Compare with Other Triangles
For a right triangle, the maximum area occurs when the triangle is isosceles (i.e., both legs are equal). However, the area of such a triangle will always be less than that of the equilateral triangle for the same circumradius.

**Hint:** Use the properties of triangles and their angles to justify why the equilateral triangle yields the maximum area.

### Conclusion
After comparing the areas of different triangles inscribed in the circle, we conclude that the triangle of maximum area that can be inscribed in a given circle of radius \( r \) is an **equilateral triangle**.

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