I've encountered a problem in which I want to maximise the lengths of the sides of a triangular structure while still fitting within an area of 220 x 100. This reduces to maximising the lengths of the sides of triangle with base 220 and height 100.

How could the maximised lengths be calculated?

Edit:- Adding clearer bounds to the problem

The triangle must fit in an area bounded by sides of 300 & 100.

  • $\begingroup$ which hight do you mean? $\endgroup$ – Dr. Sonnhard Graubner Nov 11 '17 at 12:53
  • $\begingroup$ I don't think I bounded my question well enough, I'll edit it to add more bounds. I think @user8734617 may be on the right lines though. $\endgroup$ – Era Nov 11 '17 at 13:23
  • $\begingroup$ What is the "length"? Do you mean "perimeter"? If so, use the Pythagorean theorem. $\endgroup$ – user202729 Nov 11 '17 at 13:25
  • $\begingroup$ Just consider that the distance is a convex function, the sum of convex functions is a convex function and convex functions on bounded, convex domains attain their maximum values at the boundary. $\endgroup$ – Jack D'Aurizio Nov 11 '17 at 14:07

This is not a rigorous proof, but I hope is on the right track. If you have two triangle corners fixed somewhere in the rectangle (incl. the edge), and you are looking for where to put the third one to maximise the perimeter, you are in essence trying to find the largest ellipse with those two corners as focus points, which still intersects the rectangle (and then pick one of the intersecting points). Thus, it is fairly obvious that this ellipse will catch a corner of the rectangle, i.e. the third point must be at the corner of the rectangle. By applying the same argument, you conclude that all three corners of the triangle must coincide with the corners of the rectangle. Thus, your maximum is achieved on a right-angled triangle having two sides of the rectangle as its own sides.

  • $\begingroup$ I assumed we were looking for the triangle of the largest perimeter contained in a rectangle of a given size. On second reading of the (edited) question, I see it is not clear that this is what OP is asking. $\endgroup$ – user491874 Nov 11 '17 at 13:35
  • $\begingroup$ It is the largest perimeter of a triangle within a rectangle of a given size, with the stipulation that the base is limited to a length of 220. I think, through adapting your answer, I have found the solution $\endgroup$ – Era Nov 11 '17 at 14:00

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