document.write( "Question 1210567: In cyclic quadrilateral $PQRS,$
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\n" ); document.write( "Find the largest side in quadrilateral $PQRS,$ in degrees. \n" ); document.write( "

Algebra.Com's Answer #853700 by ikleyn(53617) $\"\"$ $\"About$

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\n" ); document.write( "In cyclic quadrilateral PQRS, angle P = 30, angle Q = 60, PQ = 4, QR = 8.
\n" ); document.write( "Find the largest side in quadrilateral PQRS, in degrees.
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\n" ); document.write( "\n" ); document.write( "        Obviously, the question in the post is posed in mathematically illiterate way.\r
\n" ); document.write( "\n" ); document.write( "        But I will not pay attention to it.\r
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\n" ); document.write( "\n" ); document.write( "        My goal here is to prove that such a quadrilateral, as described in the post, \r
\n" ); document.write( "\n" ); document.write( "        DOES NOT EXIST and CAN NOT EXIST.\r
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document.write( "Let's consider triangle PRQ, formed by sides PQ and QR and the diagonal PR of quadrilateral PQRS.  \r\n" );
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document.write( "Since the quadrilateral PQRS is cyclic, it is convex, so the diagonal PR lies inside this quadrilateral.\r\n" );
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document.write( "In this triangle, we are given the side lengths  PQ = 4  and  QR = 8.\r\n" );
document.write( "We also are given the contained angle  Q  of  60°.\r\n" );
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document.write( "So, we can compute the opposite side PR of this triangle using the cosine law\r\n" );
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document.write( "    PR =  =  =  =  = .\r\n" );
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document.write( "Now we see that   +  =  +  = 16 + 48 = 64 =  = .\r\n" );
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document.write( "Hence, triangle PQR is a right-angled triangle with the legs PQ and PR with the right angle at vertex P.\r\n" );
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document.write( "But angle P is 30°, as it is given in the problem - - - so, it CAN NOT contain angle QPR of 90°.\r\n" );
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document.write( "This CONTRADICTION proves that quadrilateral PQRS as described in the post, DOES NOT EXIST and CAN NOT EXIST.\r\n" );
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document.write( "        The problem describes something geometric shape that does not exist and can not exist,\r\n" );
document.write( "        and the problem tries to implant this wrong/false idea into the reader's mind.\r\n" );
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\r
\n" ); document.write( "\n" ); document.write( "Thus the problem is refuted, i.e. killed to the death and ruined into dust.\r
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\n" ); document.write( "\n" ); document.write( "The posed problem tries to conceive a reader, as it do many other problems from the same source,
\n" ); document.write( "which I call \" the Laboratory for false defective problems in Geometry \".\r
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