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 Maximum Triangle (Posted on 2005-09-19)
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 | 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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