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Maximum Triangle (Posted on 2005-09-19) Difficulty: 3 of 5
Triangle ABC has a point D on side BC such that BA=AD=DC=1. What is angle ABD when the area of the triangle is maximized, and what is the maximum area?

See The Solution Submitted by Brian Smith    
Rating: 1.5000 (4 votes)

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Solution Solution | Comment 1 of 4
 
Since AD = AB, angle ADB = angle ABD.
Let t = angle ABD. Therefore, 0 < t < 90 and BD = 2*cos(t).

                BA*BC*sin(t)     1*(2*cos(t) + 1)*sin(t)
  Area(ABC)  = -------------- = -------------------------
                     2                       2

                (4*cos(t)^2 + cos(t) -2)                     sqrt(33) - 1
  Area(ABC)' = -------------------------- = 0  ==> cos(t) = --------------
                            2                                      8

                -sin(t)*(8*cos(t) + 1)
  Area(ABC)" = ------------------------ < 0  for 0 < t < 90  
                           2

Therefore,
  
  Angle ABD ~= 53.624808 degrees for maximum
  Area(ABC) ~=  0.880086
 

  Posted by Bractals on 2005-09-19 16:22:44
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