Geometry - semicircle inside isosceles triangle

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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....

  • Geometry - semicircle inside isosceles triangle -

    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

  • Geometry - semicircle inside isosceles triangle -

    Thanks a lot!

  • Geometry - semicircle inside isosceles triangle -

    Stop Cheating!

  • Geometry - semicircle inside isosceles triangle -

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

    However, it is still quite funny...

  • Geometry - semicircle inside isosceles triangle -

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

  • Geometry - semicircle inside isosceles triangle -

    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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