document.write( "Question 1184169: If p, q and r are the sides opposite to the angles P, Q and R respectively in a triangle PQR. If $\"+r%5E2SinPSinQ+=+pq\"$ then the triangle PQR is:
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\n" ); document.write( "(e) Right Angle as well as Isosceles \n" ); document.write( "

Algebra.Com's Answer #814698 by ikleyn(52810) $\"\"$ $\"About$

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document.write( "It is very well known fact from Geometry that for any triangle PQR with the sides p, q, r opposite to angle P,Q, R respectively,\r\n" );
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document.write( "the ratios  ,  ,    are all equal and are equal to the diameter of the circle circumscribed around the triangle PQR\r\n" );
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document.write( "      =  =  = d.\r\n" );
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document.write( "For the proof see, for example, lesson\r\n" );
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document.write( "    - Law of sines - the Geometric Proof \r\n" );
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document.write( "in this site.\r\n" );
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document.write( "Then from the given part of the problem, we see that\r\n" );
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document.write( "     = . = d*d = d^2.\r\n" );
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document.write( "In other words, the square of the side \"r\"  is equal to the square of the diameter of circumscribed circle\r\n" );
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document.write( "              = .\r\n" );
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document.write( "It implies  r = d,  i.e.  the side \"r\"  is the diameter of the circumscribed circle about triangle PQR.\r\n" );
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document.write( "It means that the angle R is leaning on the diameter of the circumscribed circle,\r\n" );
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document.write( "i.e. triangle PQR is a right-angled triangle with the right angle at vertex R.\r\n" );
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document.write( "So, the given triangle is a right-angled triangle.\r\n" );
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document.write( "ANSWER.  PQR is a right-angled triangle.\r\n" );
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