Finding an equilateral triangle inscribed inside a square of side 4

[gamaz321]

Hello,
Here we need to find the equilateral triangle that will have the maximum area. Since the equilateral triangle is inscribed inside the square is it safe to assume that the three vertices of the equilateral triangle should be touching the sides of the square, right? In any event, how does one get started here? Any clue will be very helpful.

 

[BigBeachBanana]

Find what of the triangle? The area, coordinates, perimeter? Please state the problem exactly and do not paraphrase.

 

[gamaz321]

Sorry, I missed the area word. Thank you.

 

[khansaheb]

Sorry, I missed the area word. Thank you.

Please post the corrected and total problem statement along with any sketch - as it was presented to you.

 

[Dr.Peterson]

Hello,
Here we need to find the equilateral triangle that will have the maximum area. Since the equilateral triangle is inscribed inside the square is it safe to assume that the three vertices of the equilateral triangle should be touching the sides of the square, right? In any event, how does one get started here? Any clue will be very helpful.

I see nothing missing in the problem that is not in the title. (It's better to put everything in the text of a problem.) But it's true that the exact wording may help us.

As far as assumptions, I would avoid assuming anything I am not sure of! It's conceivable (even if not likely) that only two vertices of the triangle might be on the edges of the square; and those might be on the same side, or different sides. So (without having done any work on it yet) I might start by supposing that two vertices are both on the bottom side and find the largest possible area; then suppose they are on two adjacent sides; and so on. You may discover as you go that some possibilities can be skipped with good reason.

Please show us anything at all that you have tried; and do you know only algebra and geometry, or can you use some calculus? Any clue about what you know and what you need may be helpful.

 

[gamaz321]

OK, I tried exactly as you are suggesting. I have drawn the equilateral triangle with the base of the square as one side and the other two sides making sixty-degree angles to the base to join at the intersection point. However, this leads to an equilateral triangle of side 4. The value of the area of the triangle does not match the answer.

 

[Dr.Peterson]

OK, I tried exactly as you are suggesting. I have drawn the equilateral triangle with the base of the square as one side and the other two sides making sixty-degree angles to the base to join at the intersection point. However, this leads to an equilateral triangle of side 4. The value of the area of the triangle does not match the answer.

But if that's all you did, then you haven't done "exactly as I suggested"! I said,

I might start by supposing that two vertices are both on the bottom side and find the largest possible area; then suppose they are on two adjacent sides; and so on.

Did you continue to the next step?

Please do so, and show your work.

Also, if the answer is an area, rather than a description of the triangle, then it's true that you didn't show us the entire problem, even after being asked to. Please to that, too.Also, for completeness, you should show us what answer is provided. Sometimes that can be important, as it might show what form of answer they want, or it may even be wrong, and we could check it.

 

[blamocur]

I found this diagram (drawn for the square with side 1) helpful:

 

[gamaz321]

I found this diagram (drawn for the square with side 1) helpful:

Thanks. In the diagram does it give the coordinate values of the vertices?

 

[blamocur]

Thanks. In the diagram does it give the coordinate values of the vertices?

The diagram is meant to serve as a hint, not an answer.

 

[blamocur]

The diagram is meant to serve as a hint, not an answer.

More details: in the updated diagram (attached) I wrote an equation for dependency between $u$ and $v$, then expressed the length of a side of the triangle as a function of $u$ and $v$ and figured out how to maximize it.

 

[Dr.Peterson]

Thanks. In the diagram does it give the coordinate values of the vertices?

There are many ways you might approach the problem; for instance, you might define a position for the triangle in terms of a pair of vertices (defined by any of several possible variables), or one vertex and an angle, or @blamocur's idea of specifying the midpoint of an edge and the opposite vertex, and so on. And you might use calculus all the way through, or you might use geometry or logic to decide what cases deserve attention.

An alternative I just thought of, which gave me an answer very quickly (if I trust my logic) is to turn the problem inside-out and look for the orientation of a triangle for which the greatest dimension of a bounding box (horizontal and vertical distances between leftmost and rightmost vertices) is smallest.

Once you show some sort of work, we'll be able to guide that work, or suggest a better start using the same tools.