Geometry - semicircle inside isosceles triangle

posted by Knights

Isosceles triangle ABC has sides of length AB=AC=25 and BC=40 . Find the area of a semicircle inscribed in triangle ABC with diameter along BC .

Please help I do not know how to start....

  1. Steve

    If the center of the semi-circle is O, draw OD where D is the point where AC is tangent to o the circle.

    Since OD┴AD, if the radius is r, let a=AD and b=DC, so a+b=25

    r^2 + a^2 =15^2
    r^2 + b^2 = 20^2

    equating r^2, we get
    15^2-a^2 = 20^2-(25-a)^2
    solve to get a=9, so b=16

    So, r^2+9^2 = 15^2
    r^2 = 225-81=144

    area of semi-circle is 1/2 pi r^2 = 72pi

  2. Knights

    Thanks a lot!

  3. AOPS

    Stop Cheating!

  4. Overseer

    Wow...AoPS caught you...don't know what to say...

    However, it is still quite funny...

  5. AoPS

    In violation of our honor code. The previous AoPS was an imposter, his username had an uppercase "O" instead of the correct lowercase o.

  6. Anonymous

    i agree- and please ask for hints on the message boards. THe questions are supposed to be challenging- try to solve them by yourself

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