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A semicircle has AB as the endpoints of its diameter, and AB=400. Points C and D are on the circumference of the semicircle such that AD=BC=100. What is the length of DC?

  • maths -

    let your centre be O.
    Join OD and OC
    Triangles AOD and BOD are clearly congruent (SSS)

    let's look at triangle BOD, with sides 200, 200 , 100
    and its central angle BOD = Ø
    A "similar" triangle 2 , 2, 1 has the same angle Ø
    by cosine law:
    1^2 = 2^2 + 2^2 - 2(2)(2)cosØ
    8cosØ = 7
    cosØ = 7/8

    It can be easily seen that CD || AB
    so in triangle COD we have angle ODC = Ø
    Let CD = 2x, draw an altitude from O to meet CD at P
    clearly P is the midpoint, OPD is right-angled and
    x/200 = cosØ
    x = 200cosØ = 200(7/8) = 175
    then 2x or CD = 350

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