Algebra.Com's Answer #851784 by ikleyn(52776)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "A parallelogram has diagonals 34 in and 20 in and one side measures 15 in.
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document.write( "(a) Find the length of the other side
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document.write( "(b) Find the area
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document.write( "(c) Find the largest interior angle of the parallelogram
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document.write( " (a) Find the length of the other side\r\n" );
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document.write( "The diagonals of a parallelogram bisect each other and divide the parallelogram \r\n" );
document.write( "in 4 (four) small triangles.\r\n" );
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document.write( "Let's consider one of four small triangles, formed by two intersecting diagonals and the given side.\r\n" );
document.write( "This triangle has the sides of 34/2 = 17 in, 20/2 = 10 in and 15 in.\r\n" );
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document.write( "Write the cosine law for this triangle to find the cosine of the angle 'a' between the diagonals\r\n" );
document.write( "opposite to the side of 15 inches long\r\n" );
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document.write( " 15^2 = 17^2 + 10^2 - 2*17*10*cos(a),\r\n" );
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document.write( " cos(a) =  =  =  .\r\n" );
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document.write( "The other angle between the diagonals, 'b', is supplementary to angle 'a'. \r\n" );
document.write( "Hence, angle 'b' has the cosine  .\r\n" );
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document.write( "Therefore, the square of the side of the parallelogram, opposite to angle 'b', according the cosine law, is\r\n" );
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document.write( " 17^2 + 10^2 - 2*17*10*cos(b) =  = 17^2 + 10^2 + 2*82 = 553.\r\n" );
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document.write( "Hence, the opposite side of the parallelogram to angle 'b' is  = 23.51595203.\r\n" );
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document.write( " It is the ANSWER to question (a).\r\n" );
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document.write( " (b) Find the area\r\n" );
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document.write( "For any parallelogram, its diagonals bisect each other\r\n" );
document.write( "and divide a parallelogram in 4 (four) small triangles.\r\n" );
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document.write( "So, one of such triangles has the sides 34/2 = 17 inches, 20/2 = 10 inches\r\n" );
document.write( "and 15 inches.\r\n" );
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document.write( "Having three sides of this triangle, we can find its area using the Heron's formula.\r\n" );
document.write( "For shortness, I will not write the formulas, since they are in each textbook.\r\n" );
document.write( "I simply will use one of many existing online calculators for it.\r\n" );
document.write( "So, the area of this specific triangle is \r\n" );
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document.write( "  = 74.458 in^2.\r\n" );
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document.write( "The entire parallelogram is the union of small triangles.\r\n" );
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document.write( "The interesting fact is that the areas of all these triangles are equal.\r\n" );
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document.write( "You can easy prove it for yourself, if you draw a perpendicular from one of the vertex\r\n" );
document.write( "of the parallelogram to the opposite diagonal. This perpendicular then will be the common\r\n" );
document.write( "altitude of two small triangles. The bases of these small triangle are equal\r\n" );
document.write( "and the altitude is common - so, the areas of these triangles are the same.\r\n" );
document.write( "The similar proof works for the opposite vertex and two other triangles.\r\n" );
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document.write( "Thus the area of the parallelogram is 4 times the area of any of small triangles\r\n" );
document.write( "of the subdivision. Thus the area of the parallelogram is  = 297.832 in^2.\r\n" );
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document.write( " It is the ANSWER to question (b).\r\n" );
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document.write( " (c) Find the largest interior angle of the parallelogram\r\n" );
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document.write( "Let 'C' be the largest interior angle of the parallelogram.\r\n" );
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document.write( "For any two adjacent sides s1 and s2 of a parallelogram, the area of the parallelogram is\r\n" );
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document.write( " area = s1*s2*sin(C), where C is the angle concluded between these sides.\r\n" );
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document.write( "We just found the area of the parallelogram in section )b) above and we know that the area is 297.832 in^2.\r\n" );
document.write( "So, we can write this equation\r\n" );
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document.write( " 297.832 = 15*23.51595203*sin(C).\r\n" );
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document.write( "It gives sin(C) =  = 0.844340329.\r\n" );
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document.write( "Since 'C' is the largest interior angle, it is obtuse angle, so we can write\r\n" );
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document.write( " C = 180° - arcsin(0.844340329) = 180° - 57.6013263° = 122.3986737°.\r\n" );
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document.write( "Thus the greatest angle of the parallelogram is about 122.399°.\r\n" );
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document.write( " It is the ANSWER to question (c).\r\n" );
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document.write( "Question (c) can be answered/solved in other way.\r\n" );
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document.write( "Write the cosine law for angle C and the opposite to it longest diagonal of 34 inches\r\n" );
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document.write( " 34^2 = 15^2 + 23.51595203^2 - 2*15*23.51595203*cos(C),\r\n" );
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document.write( " cos(C) =  = -0.535806502.\r\n" );
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document.write( "Hence, \r\n" );
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document.write( " C = arccos(-0.535806502) = 122.399°,\r\n" );
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document.write( "which is consistent with we had above.\r\n" );
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document.write( "At this point, the problem is solved in full.\r
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