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A Square, a Rectangle and a Triangle (Posted on 2013-11-17) Difficulty: 3 of 5
Find the dimensions and orientation of the largest area (a) square and (b) rectangle you can draw inside an equilateral triangle of side 1

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 3.0000 (1 votes)

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disagree | Comment 2 of 3 |
I took rectangle to mean non-square.

Square

A square side=x aligned with one side on the base of the triangle and the other corners touching the other sides gives a 30-60-90 triangle with the vertical side=sqrt(3)*(1-x)/2.  But this side=x and x = sqrt(3)/(2 + sqrt(3)) = 2*sqrt(3) - 3 and area =  .2153903091+

Rectangle.

x=width, y=heighth.  Using the same 30-60-90 triangle, y = sqrt(3)*(1-x)/2 and area xy = x*sqrt(3)*(1-x)/2 which is minimum when x = 1/2, so y=sqrt(3)/4, area = sqrt(3)/8 = .2165063509+     

  Posted by xdog on 2013-11-18 19:33:32
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