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document.write( "The diagonals of a parallelogram measure 16 cm and 24 cm. The shorter side measures 10 cm.
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document.write( "1. Find the area of the parallelogram.
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document.write( "2. Find the measure of the longer side.
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document.write( "3. Find the measure of the smaller angle of the parallelogram.
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document.write( "Diagonals of a parallelogram bisect each other.\r\n" );
document.write( "They divide a parallelogram in 4 (four) small triangles.\r\n" );
document.write( "The angles between the diagonals are vertical and supplementary.\r\n" );
document.write( "So, if 
is an acute angle between the diagonals, \r\n" );
document.write( "then the other, supplementary angle 
is 
.\r\n" );
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document.write( "Let's find an acute angle between the diagonals for the given parallelogram.\r\n" );
document.write( "Obviously, this angle is opposite to the shorter side of the parallelogram.\r\n" );
document.write( "The halves of diagonals are 16/2 = 8 cm and 24/2 = 12 cm long, and acute angle \r\n" );
document.write( "is a concluded angle between the halves of diagonals.\r\n" );
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document.write( "Write the cosine law for such a triangle\r\n" );
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document.write( " 10^2 = 
+ 
- 
.\r\n" );
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document.write( "It gives\r\n" );
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document.write( " 
= 
= 
= 
.\r\n" );
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document.write( "These four small triangles of the subdivision have a remarkable property:\r\n" );
document.write( "their areas are all the same.\r\n" );
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document.write( "It is easy to prove: the area of each such a triangle is half the product of the sides\r\n" );
document.write( "that are halves of diagonals, by the sine of the angle between them.\r\n" );
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document.write( " area = 
for the acute angle between diagonals\r\n" );
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document.write( "and \r\n" );
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document.write( " area = 
for the obtuse angle between diagonals.\r\n" );
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document.write( "But the acute angle and the obtuse angle are SUPPLEMENTARY, so, they have the same value of sine.\r\n" );
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document.write( "GREAT !\r\n" );
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document.write( "Let's calculate the sine of an angle between the diagonals.\r\n" );
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document.write( "For the acute angle, it is \r\n" );
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document.write( " 
= 
= 
= 
= 
= 
.\r\n" );
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document.write( "For the obtuse angle, the sine value is the same, .\r\n" );
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document.write( "So, each of the four small triangle of the subdivision has the area\r\n" );
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document.write( " = 
= 
.\r\n" );
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document.write( "The area of the whole parallelogram is then four times the area of each separate small triangle \r\n" );
document.write( "of the subdivision.\r\n" );
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document.write( "So, the area of the parallelogram is 
= 
, or about 158.7450787 cm^2.\r\n" );
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document.write( "To find the measure of the longer side, apply the cosine law again.\r\n" );
document.write( "Use the obtuse-angled triangle of the subdivision.\r\n" );
document.write( "Its sides are 8 cm and 12 cm; the cosine of the obtuse concluded angle = 
\r\n" );
document.write( "is 
= 
.\r\n" );
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document.write( "So, we write for the square of the longer side of the parallelogram\r\n" );
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document.write( " 
= 64+144+108 = 316.\r\n" );
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document.write( "Thus the longer side of the parallelogram is 
= 17.77638883 cm (approximately).\r\n" );
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document.write( "The last question is to find the smaller angle of the parallelogram, 
.\r\n" );
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document.write( "Let's find it, using the shorter side of 10 cm and what we just learned about this parallelogram: \r\n" );
document.write( "its area is and its longer side is .\r\n" );
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document.write( "The area of any parallelogram is the product of any two adjacent sides by the sine of the concluded angle.\r\n" );
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document.write( "So, for 
we can write this equation\r\n" );
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document.write( " 
} = .\r\n" );
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document.write( "Thus = 
= 
= 0.893010837.\r\n" );
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document.write( "Hence, = arcsin(0.893010837) = 63.2540602 degrees.\r\n" );
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document.write( "ANSWER. The area of the parallelogram is , or about 158.7450787 cm^2.\r\n" );
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document.write( " The measure of the longer side is , or 17.77638883 cm (approximately).\r\n" );
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document.write( " The measure of the smaller angle of the parallelogram is arcsin(0.893010837), or about 63.2540602 degrees.\r\n" );
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document.write( "Solved.\r
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