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March 30, 2017

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Convex quadrilateral ABCD has sides AB=BC=21, CD=15 and AD=9. Given additionally that ∠ABC=60∘, what is the length of BD?

  • Geometry - ,

    Join the diagonals
    Since B - 60° and AB = BC = 21
    ABC must be equilateral and AC = 21

    Now in triangle ACD we have 3 sides, and by the cosine law I found angle D to be 120°
    interesting!
    using the sine law in triangle ACD
    sin CAD/15 = sin60/21
    sin CAD = .61858..
    angle CAD = 38.3232... (I stored that)
    angle BAD = 60 + 38.32... = 98.21..°

    by the cosine law:
    BD^2 = 21^2 + 9^2 - 2(21)(9)cos BAD
    = 528
    BD = √528 or appr 22.98

  • Geometry - ,

    Reiny in the last step u have done an error as 21^2 =441 + 9^2 = 81 = 522
    and 2*21*9*cos(98) = - 54.6
    so 522 - (-54.6)=522+52.6=574.6
    and square root of 574.6 = 23.97 ~ 24

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