Algebra.Com's Answer #848723 by ikleyn(52776)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "An isosceles triangle ABC, in which AB = BC = 6√2 and AC = 12 is folded along the altitude BD
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document.write( "so that planes ABD and BDC form a right dihedral angle. Find the angle between side AB and its
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document.write( "new position.
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document.write( " The solution by the other tutor (which, I believe, is an AI solution) is FATALLY WRONG.
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document.write( " I came to bring a correct solution.\r
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document.write( "In the isosceles triangle ABC\r\n" );
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document.write( " AB^2 + BC^2 =  + = 36*2 + 36*2 = 72 + 72 = 144\r\n" );
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document.write( "and\r\n" );
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document.write( " AC^2 =  = 144.\r\n" );
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document.write( "It means that triangle ABC is a right-angled isosceles triangle with the right angle at B,\r\n" );
document.write( "hypotenuse BC and the legs AB and BC.\r\n" );
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document.write( "Now imagine that the triangle ABC is folded along the altitude BD so the planes ABD and BDC form a right dihedral angle.\r\n" );
document.write( "Let plane BDC remains in its original position and vertex A takes new position A' in space.\r\n" );
document.write( "Then the segments A'D and CD form the right angle.\r\n" );
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document.write( "Consider triangle A'DC. It is a right-angled triangle with the right angle D and the legs A'D and CD.\r\n" );
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document.write( "These legs have equal length A'D = CD = 12/2 = 6, A'D= AD and AD and CD are equal halves of the base AC of the triangle ABC.\r\n" );
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document.write( "Therefore, the hypotenuse A'C of the triangle ADC has the length  =  =  .\r\n" );
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document.write( "Thus the triangle A'BC (after folding) has congruent sides A'B, A'C and BC: the length of each of these sides is .\r\n" );
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document.write( "Thus the triangle A'BC is an equilateral triangle and all its angles have the measure of 60°.\r\n" );
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document.write( "In particular, the angle A'BC is 60°.\r\n" );
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document.write( "The angle between AB and its new position A'B is the same 60°. The proof is the same.\r\n" );
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document.write( "Solved.\r
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