Anonymous
Anonymous asked in Science & MathematicsMathematics · 4 years ago

 AE = ... .............?

             A●

            ○

              ○ ○

B●       D●   E●   C●

△ABC,

∠BAD = ∠DAE = ∠EAC,

BD = 2,

DE = EC = 1.

 AE = ...

1 Answer

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  • atsuo
    Lv 6
    4 years ago
    Favorite Answer

    Let ∠BAD = ∠DAE = ∠EAC = θ .

    Use the sine law ,

    EC / sinθ = 1 / sinθ = AE / sin(∠ACE)

    DE / sinθ = 1 / sinθ = AE / sin(∠ADE)

    So

    sin(∠ACE) = sin(∠ADE)

    Both angles are smaller than 90° , so

    ∠ACE = ∠ADE

    Therefore △ACD is an isosceles with AC = AD , so

    ∠AEC = ∠AED = 90° .

    It means tan(2θ) = BE / AE = 3 / AE , so

    2tanθ / (1 - tan^2(θ)) = 3 / AE

    And tanθ = EC / AE = 1 / AE , so

    2(1 / AE) / (1 - (1 / AE)^2) = 3 / AE

    2AE / (AE^2 - 1) = 3 / AE

    3(AE^2 - 1) = 2AE^2

    AE^2 = 3

    AE = √3 <--- The answer

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