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 Measure that angle V (Posted on 2005-10-24)
A certain triangle ABC has D on AB and E on AC such that AD=DB=BC=CE and AE=ED. What is the measure of angle BAC?

 See The Solution Submitted by Brian Smith No Rating

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 Solution | Comment 6 of 7 |
` `
`Let x = AD = DB = BC = CE, y = AE = ED, and    z = measure of angle BAC.`
`Applying the cosine rule to triangle DAE,`
`    y^2 = x^2 + y^2 - 2xy cos(z)`
`                 or`
`    x = 2y cos(z)                                   (1)                   `
`Applying the cosine rule to triangle BAC,`
`    x^2 = (2x)^2 + (x+y)^2 - 2(2x)(x+y) cos(z)`
`                 or`
`    0 = 4x^2 + 2xy + y^2 - 4x(x+y) cos(z)           (2)`
`Substituting x from (1) into (2) we get`
`    16 cos(z)^3 - 8 cos(z)^2 - 4 cos(z) - 1 = 0`
`Solving for z using MathCad 12 we get`
`    z ~= 29.5467128538 degrees`
` `

 Posted by Bractals on 2005-10-24 20:48:48

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