Equilateral Triangle Inside A Square

Equilateral Triangle Within a Square - Problem With Solution

A trouble on the proof of an equilateral triangle inside a square is presented along with detailed solution.

Trouble

In the figure beneath, ABCD is a square. Bear witness that triangle DMC is equilateral.
Solution to Problem :

• AMB is an isosceles triangle and therefore MA = MB. Also the sizes of angles MAD and MBC are equal. Hence triangles DAM and CBM have an equal angle between two equal sides are therefore congruent. Since triangles DAM and CBM are congruent sides DM and CM are equal in size and therefore triangle DCM is isosceles.
  
  
• Nosotros draw the perpendicular MM' to CB and we consider the right triangles MCM' and MBM' and the tangent of angles MCM' and MBM'.
  tan(MCM') = MM' / CM'
  tan(MBM') = MM' / BM'
  
• The above may exist written as
  CM' = MM' / tan(MCM') and BM' = MM' / tan(MBM')
  
• Note that the size of bending MBM' is equal to 75 degrees. Let the length of the side of the square be x. Hence
  MM' = x / 2 and CM' + BM' = x
  
• Substitute the above in the equations CM' = MM' / tan(MCM') and BM' = MM' / tan(MBM') to obtain
  10 = (x/2) / tan(MCM') + (x/two) / tan(75 degrees)
  
• Split all terms of the equation by x and multiply them past 2 to obtain
  2 = 1 / tan(MCM') + one / tan(75 degrees)
  
• tan(75 degrees) is calculated using the tangent of a sum formula follows
  tan(75 degrees) = tan(thirty degrees + 45 degrees) =
  [ tan(thirty) + tan(45) ] / [ 1 - tan(30)*(tan(45) ] = 2 + sqrt(3)
  
• Substitute tan(75 degrees) by ii + sqrt(three) into the equation two = 1 / tan(MCM') + i / tan(75 degrees) to obtain
  1 / tan(MCM') = 2 - 1 / [ 2 + sqrt(three) ]
  
• Simplify the above and solve for tan(MCM') to obtain
  tan(MCM') = 1 / sqrt(3)
  
• Solve the above trigonometric eqution to detect the size of bending MCM'
  MCM' = thirty degrees
  
• The size of angle DCM is now calculated every bit follows
  DCM= xc degrees - 30 degrees = threescore degrees
  
• and since triangle DCM is isosceles then angle CDM is as well equal to lx degrees and the triangle is equilateral.

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Equilateral Triangle Inside A Square,

Source: https://www.analyzemath.com/Geometry/challenge/equilateral_square.html

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