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Math Help - Circles area proving

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    Circles area proving

    The diagram shows a rectangle abcd inscribed in a circle. Given AB=4x cm and BC=8 cm, show that the area, P cm^2 of the shaded region is given by P = 4(\pi x^2+4 \pi -8x)

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    [quote=Punch;527303]The diagram shows a rectangle abcd inscribed in a circle. Given AB=4x cm and BC=8 cm, show that the area, P cm^2 of the shaded region is given by P = 4(\pi x^2+4 \pi -8x)

    You can work out the diameter using the Pythagorean Theorem.

    (4x)^2 + 8^2 = D^2

    16x^2 + 64 = D^2

    D = \sqrt{16x^2 + 64}.


    Therefore the radius r = \frac{\sqrt{16x^2 + 64}}{2}.


    So the area of the circle is

    A_{\textrm{Circle}} = \pi r^2

     = \pi \left(\frac{\sqrt{16x^2 + 64}}{2}\right)^2

     = \pi \left(\frac{16x^2 + 64}{4}\right)

     = \pi (4x^2 + 16)

    4\pi x^2 + 16\pi.


    Now if we work out the area of the rectangle,

    A_{\textrm{Rectangle}} = 4x \cdot 8

     = 32x.



    So the area of the shaded region is

    A_{\textrm{Shaded}} = A_{\textrm{Circle}} - A_{\textrm{Rectangle}}

     = 4\pi x^2 + 16\pi - 32x

    = 4(\pi x^2 + 4\pi - 8x).
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