Proofs q (1 Viewer)

[kkk579]

Im a little confused by this question - how exactly is the solution proving that the maximum area occurs when the triangle is equilateral?

 

[kkk579]

Also what types of proofs qs come in frequently in the hsc?

 

[kkk579]

bump

 

[kendricklamarlover101]

equality is achieved when x=y=z from the 2nd line and that holds for the rest of the proof

 

[kkk579]

equality is achieved when x=y=z from the 2nd line and that holds for the rest of the proof

Oh so you dont directly use that info in ur proof or derive it from the final step of ur proof? U js state when rquality pccurs?

 

[kendricklamarlover101]

Oh so you dont directly use that info in ur proof or derive it from the final step of ur proof? U js state when rquality pccurs?

in the final line u proved that the area is less than a value, which means that value is the maximum. since equality occurs when x=y=z, then the maximum occurs when it is an equilateral triangle

 

[kkk579]

in the final line u proved that the area is less than a value, which means that value is the maximum. since equality occurs when x=y=z, then the maximum occurs when it is an equilateral triangle

Ohhhhh

 

[kkk579]

Ohhhhh

Okay yeah that makes sense

 

[Luukas.2]

It's worth noting that this proof would not have established the maximum area at all without the equality part.

Leaving out the equality part, the proof establishes that the area of any triangle with semi-perimeter s cannot be more than s² / root 27, but that doesn't mean that a triangle with that area is possible. It is like my saying that a circle of radius 1 fits entirely with a square of side length 4, and so the area of the circle is at most 4 square units - the proof establishes an upper bound. It is only the equality part that shows that a triangle of the stated area does exist and that it is equilateral. The proof does not formally address that any non-equilateral triangle with that perimeter will have a smaller area, though that is true from the AM-GM as equality occurs if, and only if, the terms are all equal.

 

[StudyNotesTips]

1. Understanding the Upper Bound

• Upper Bound Defined: The proof you referenced establishes that the area AAA of any triangle with a semi-perimeter sss is at most s227\frac{s^2}{\sqrt{27}}27s2. This means no triangle with that semi-perimeter can have an area greater than this value.
• Importance of Bounds: While upper bounds can be useful for understanding the limits of a system, they do not provide complete information about possible configurations. Just knowing the upper limit doesn’t guarantee that triangles exist that can reach that limit.

2. Role of the Equality Condition

• Equality Condition: The equality condition is essential for concluding that an equilateral triangle exists with the maximum area. This condition often appears in proofs involving inequalities (like AM-GM) and indicates when the maximum is achieved.
• Significance of Equilateral Triangles: In this case, the equilateral triangle is unique because it distributes its side lengths evenly, maximizing the area for a given perimeter. Any deviation from equal sides (i.e., forming a non-equilateral triangle) will lead to a smaller area.

3. Implications of the AM-GM Inequality

• AM-GM Context: The Arithmetic Mean-Geometric Mean (AM-GM) inequality states that for non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean, with equality holding when all numbers are equal.
• Application in Geometry: When applying AM-GM to the triangle's side lengths, it shows that only an equilateral triangle can yield the area equal to the upper bound because only in this case are all side lengths equal.

4. Geometric Intuition

• Visualizing the Maximum: If you visualize the triangle and its area, the equilateral triangle not only maximizes the area for the given semi-perimeter but also maintains the symmetry that often leads to optimal configurations in geometry.
• Area Comparisons: If you consider various triangle shapes (isosceles, scalene), you can see how variations in angles and side lengths result in reduced areas compared to the equilateral triangle for the same semi-perimeter.

5. Generalization to Other Shapes

• Comparison with Other Geometric Shapes: Similar principles apply in other geometric contexts, such as the comparison of circles to polygons. For instance, a circle maximizes area relative to its perimeter compared to any polygon, much like how the equilateral triangle maximizes area among triangles.

6. Proof Structure and Rigor

• Proof Completeness: For a proof to be comprehensive, it should not only provide an upper limit but also address the conditions under which that limit can be reached. In mathematics, this rigor is essential for establishing clear and accurate conclusions.
• Encouraging Deeper Exploration: This discussion encourages further exploration into other types of inequalities and their implications in different geometric contexts, fostering a deeper understanding of mathematical relationships.

Hope this helps