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Chord Harmonics (Posted on 2005-01-11) Difficulty: 3 of 5
Four chords of a circle are arranged in such a way that they form a quadrilateral shape, the length of the chords are as follows.

6cm, 7cm, 8cm and 9cm.

what is the area of the circle that will intersect all of the points?

No Solution Yet Submitted by Juggler    
Rating: 4.0000 (3 votes)

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Solution Solution Comment 9 of 9 |

The endpoints of each chord are the vertices of a cyclic quadrilateral and we are to find the area of the circumcircle of the cyclic quadrilateral.

The radius of the circumcircle is given by the formula:
R = 1/4*SQRT(((ac+bd)(ad+bc)(ab+cd))/((s-a)(s-b)(s-c)(s-d))),
where a, b, c and d are the side lengths of the cyclic quadrilateral and s is the semiperimeter, i.e., (a+b+c+d)/2.
Therefore, R = 1/8*SQRT(38665/21)) =~ 5.363636937 cm, and thus, the area of the circumcircle, using the well known equation pi*Ris approximately 90.37922615 cm2

  Posted by Dej Mar on 2008-03-02 08:06:26
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