document.write( "Question 261230: if one side of a triangle is double the other and the angles opposite these sides differ by 60. show that the triangle is right angled \n" ); document.write( "

Algebra.Com's Answer #192419 by drk(1908) $\"\"$ $\"About$

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Let A, B, C be the sides and y, y-60 be the angles.
\n" ); document.write( "we know
\n" ); document.write( "(1) side A = x, side C = 2x
\n" ); document.write( "(2) angle a = y-60, angle c = y
\n" ); document.write( "--
\n" ); document.write( "using the fact that all angles add to 180, we know that 180 - y - (y-60) = angle b, which is
\n" ); document.write( "angle b = 240 - 2y.
\n" ); document.write( "--
\n" ); document.write( "using law of sines we get
\n" ); document.write( "(i) $\"%282x%2Fsiny%29+=+%28x%2Fsin%28y-60%29%29\"$
\n" ); document.write( "cross multiply to get
\n" ); document.write( "(ii) $\"2xsin%28y-60%29+=+xsin%28y%29\"$
\n" ); document.write( "use the sin subtraction identity to get
\n" ); document.write( "(iii) $\"2x%2A%28sin%28y%29cos%2860%29+-+cos%28y%29+sin%2860%29%29+=+xsin%28y%29\"$
\n" ); document.write( "simplify to get
\n" ); document.write( "(iv) $\"xsin%28y%29+-+%28sqrt%283%29%2F2%29cos%28y%29+=+xsin%28y%29\"$
\n" ); document.write( "subtract to get
\n" ); document.write( "(v) $\"-+%28sqrt%283%29%2F2%29cos%28y%29+=+0\"$
\n" ); document.write( "this means that cos(y) = 0
\n" ); document.write( "and y = 90 or 180.
\n" ); document.write( "SInce y cannot be 180 for a triangle, we have
\n" ); document.write( "y = 90.
\n" ); document.write( "This tells us that the triangle MUST be right. \n" ); document.write( "

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