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Math Help - Square outiside circle outside quadrilateral

  1. #1
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    Square outiside circle outside quadrilateral

    This was a problem on my brothers maths test, and it is doing my families collective heads in.

    A quadrilateral with three sides measuring 15cm and one side 20cm sits within a circle with each of its four corners touching the circle.
    Similarly, a square sits around the outside of the circle, touching in the middle of each side. What is the area of the square?

    Also, the problem had to be solved without a calculator.
    Last edited by Linus; September 19th 2012 at 04:45 PM. Reason: Additional info
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  2. #2
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    Re: Square outiside circle outside quadrilateral

    see attached diagram ...

    1. trapezoid diagonal bisects the base angle

    2. notice small right triangle formed by trapezoid height ... \cos(2x) = \frac{2.5}{15} = \frac{1}{6}

    3. inscribed angle is half the central angle

    4. using law of cosines ...

    15^2 = r^2 + r^2 - 2r^2\cos(2x) = 2r^2[1 - \cos(2x)]

    2r^2 = \frac{15^2}{1 - \cos(2x)}

    2r^2 = \frac{15^2}{1 - \frac{1}{6}}

    2r^2 = \frac{6 \cdot 15^2}{5}

    5. area of large square (not drawn) is 4r^2 ...

    4r^2 = \frac{2 \cdot 6 \cdot 15^2}{5} = 540
    Attached Thumbnails Attached Thumbnails Square outiside circle outside quadrilateral-trapproblem.png  
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  3. #3
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    Re: Square outiside circle outside quadrilateral

    I constructed the regular trapezoid and found the radius of the circumcircle to be 11.5 cm. side of square 23cm. area of square 529 cm^2. No calculator.
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    Re: Square outiside circle outside quadrilateral

    Quote Originally Posted by bjhopper View Post
    I constructed the regular trapezoid and found the radius of the circumcircle to be 11.5 cm. side of square 23cm. area of square 529 cm^2. No calculator.
    a 2% error ... not bad.

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    Re: Square outiside circle outside quadrilateral

    thanks Skeeter, test was without a calculator, did you manage that?
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    Re: Square outiside circle outside quadrilateral

    thanks BJ, we did that too on the computer in publisher (? i know) and got a diameter of 23.306, area of square 543.16963. Still, test was without rulers or calculator. Test was actually an Australia wide maths comp for year 10 students.
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  7. #7
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    Re: Square outiside circle outside quadrilateral

    Quote Originally Posted by Linus View Post
    thanks Skeeter, test was without a calculator, did you manage that?
    did you read my solution? no calculator required.
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