For a given perimeter (or a fixed sum of sides) of a triangle, the area is maximized when the triangle is equilateral.

Explanation:

1. Given: The sum of the sides of the triangle is fixed (i.e., the perimeter is constant).

2. To Maximize Area: The area $A$ of a triangle with sides $a, b,$ and $c$ can be expressed using Heron's formula:

   $A = \sqrt{s(s - a)(s - b)(s - c)},$

   where $s$ is the semi-perimeter of the triangle:

   $s = \frac{a + b + c}{2}.$

3. Why Equilateral Triangle?:

   • When the triangle is equilateral (i.e., $a = b = c$), the expression under the square root in Heron's formula is maximized.
   • To see this, observe that Heron's formula achieves its maximum value when the products $(s - a), (s - b), (s - c)$ are equal, which occurs only when $a = b = c$.

4. Conclusion:

   For a triangle with a fixed sum of sides (fixed perimeter), the area is maximized when all three sides are equal, forming an equilateral triangle.

Would you like more details or have any questions?

Additional Questions to Explore:

1. How do we derive Heron's formula for the area of a triangle?
2. Can you find the area of an equilateral triangle with a given perimeter?
3. How does the area change if we only know two sides and an included angle?
4. What is the effect of fixing one side of a triangle and varying the other sides on the area?
5. What other geometric shapes have maximum areas for given perimeters?

Tip: For optimization problems in geometry, symmetry often plays a key role in maximizing or minimizing quantities!