AB is a diameter of a circle and C is any point on the circumference of the circle. Then

Options

A. the area of `Delta`ABC is maximum when it is isosceles.
B. the area of `Delta`ABC is minimum when it is isosceles.
C. the perimeter of `Delta`ABC is maximum when it is isosceles.
D. none of these

Answer

To solve the problem, we will analyze the situation where AB is the diameter of a circle and C is any point on the circumference of the circle. We will determine the conditions under which the area of triangle ABC is maximized or minimized.

### Step-by-Step Solution:

1. **Understanding the Circle and Triangle**:
- Let AB be the diameter of the circle.
- Let O be the center of the circle.
- Point C is any point on the circumference of the circle.

2. **Identifying the Triangle**:
- The triangle we are considering is triangle ABC, where A and B are the endpoints of the diameter, and C is a point on the circumference.

3. **Area of Triangle**:
- The area \( A \) of triangle ABC can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
- Here, the base is the length of the diameter AB, and the height is the perpendicular distance from point C to line AB.

4. **Base of the Triangle**:
- The length of the diameter AB is constant, as it is a fixed line segment in the circle.

5. **Height of the Triangle**:
- The height of the triangle varies depending on the position of point C on the circumference. The maximum height occurs when point C is directly above the center O of the circle.

6. **Maximum Area Condition**:
- The area of triangle ABC is maximized when the height (OC) is at its maximum. This happens when C is at the highest point on the circumference, directly above O.
- In this case, triangle ABC becomes an isosceles triangle (AC = BC), and its area is maximized.

7. **Minimum Area Condition**:
- The area of triangle ABC is minimized when point C coincides with either point A or point B. In this case, the height becomes zero, leading to an area of zero.
- Thus, triangle ABC cannot form an isosceles triangle in this scenario.

8. **Conclusion**:
- The area of triangle ABC is maximum when C is at the highest point on the circumference, forming an isosceles triangle.
- The area is minimum (zero) when C is at points A or B.

### Final Answer:
- The area of triangle ABC is maximum when it is an isosceles triangle.