Hi,
I wanted to confirm whether my interpretation is correct or not. Could someone please confirm? Is there any other caveat to this? If yes, I would be grateful if you could explain.
No, when it is equilateral?
I also thought the same but apparently a lot of sites on google claim otherwise-
It says that if we are given two sides, Maximum area occurs when it is a right angled triangle. For the whole perimeter, it is equilateral triangle. I am pretty confused about this differentiation.
These are right
I still did not get you because if all three sides are fixed, so is the area and perimeter.
The rule states that only 2 sides are fixed. For instance, if the sides measure 6 and 8, the third side can range anywhere between 2 and 14. According to this rule, to maximize the area of the triangle given two sides, the optimal scenario occurs when it forms a right-angled triangle. This means that the area will be largest when the third side is 10 (we have this flexibility because there are no perimeter restrictions). However, if there were a perimeter restriction, we would opt for an equivalent triangle.
This is my interpretation of this rule but I wanted confirmation on the reasoning that’s why I asked
There’s no “reasoning” here lol bc you just stated or made an assertion and called it a rule. Nonetheless, what you said is correct as mentioned above.
It doesn’t have to be a “rule.” The area of a triangle is, as you might know, formulated as A = \frac 12 bh.
If you’re given only two sides and no other information, you could try to “fix” one side as the base (treat it as a constraint of sorts) and treat the other side as the height (why?). It’s clear now that maximizing the height thus maximizes the area of the said triangle. Our goal now then is to maximize the height.
Owing to the above “statement”, you’ll realize the height is maximized when it is perpendicular to the fixed base (see the image and caption below).
Whatever the more important thing is the idea right, not sure why you are being so particular about the terms lmao. There are proofs for the same that’s why I referred it that way.
I’m not being particular about it. I just don’t think this is something that important where you have to call it a “rule”. Maybe a change in mindset of like what you view as a rule and what you don’t might be helpful as well.
After all, it’s good to have an idea of what’s “important” and what isn’t, right?
This concept came up in a question, and I realized I was unfamiliar with it. However, after reading about it, I found it to be something new and important. You are absolutely right—it’s valuable to distinguish between what is a ‘rule’, what is important and what isn’t. That said, I think there might be better ways to communicate it. For many of us, including myself, returning to math after a long time can feel intimidating, especially when discussing concepts we’re not entirely confident about. That’s all I wanted to share but I also acknowledge how diligently you share your insights about everything so thank you for that 
I wasn’t being pedantic about your “vocab”, but rather what u deemed as a “rule” or not. The main point of my message was to clarify that this isn’t something important to the point that u have to refer to it as a rule. The intent behind that is to help you distinguish between things that matter and things that don’t.
