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      Geometry through folding Assignment

      Return here Step I Individual part  of the exercise

       Goal: Find an equilateral triangle which is maximal in the sense that it has the biggest  possible area among all equilateral triangles inside a fixed square. You can follow the steps given below or discover your own way. Explain in either case, why your method works.

      1.

      a)     Fold a triangle with angles 30 degree, 60 degree and 90 degree from your square.

              Hint: The longest edge (hypotenuse) must then be twice as long as one of the sides.

       

      b)    Make use of your observation in a)  to produce an equilateral triangle with side length equal to your original square.

       

      c)      Can you find an equilateral triangle, whose area is bigger than the one above, inside your square?

       

      2.

      a)     If a triangle is maximal, does it mean a restriction to assume that one of its vertices is in a vertex of the square? Why?

       

      b)    Assume that one of the vertices of the triangle is at the lower left vertex of the square. Denote by Q the angle between the bottom edge of the square and the bottom edge of the triangle. Write down the area of the triangle in terms of Q and find the maximal triangle. Assume the side length of the square is 1.

       

      c)     You may also suggest other ways to find a maximal triangle.

      d)    Find a folding method that produces this triangle.

       

      Follow up question (voluntary):

      Find the largest regular hexagon inside a square. 

      Hint: The previous exercise can be utilized here.

       

      Return your own solution to My Courses by 21st March


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