document.write( "Question 646150: What is the length of the diagonals of a rhombus with side = 5cm and area = 24cm^2? \n" ); document.write( "

Algebra.Com's Answer #405743 by Edwin McCravy(20055) $\"\"$ $\"About$

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What is the length of the diagonals of a rhombus with side = 5cm and area = 24cm^2?
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document.write( "The rhombus is made of 4 right triangles. The base of each of those\r\n" );
document.write( "right triangles is x and the height of each one is y.\r\n" );
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document.write( "The area of each of those 4 right triangles is\r\n" );
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document.write( "        A = ·b·h\r\n" );
document.write( "        A = ·x·y\r\n" );
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document.write( "So the area of the whole rhombus is 4 times that.\r\n" );
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document.write( "Area of rhombus = 4··x·y = 2xy\r\n" );
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document.write( "We are given that the area of the rhombus is 24 cm², so\r\n" );
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document.write( "     2xy = 24\r\n" );
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document.write( "By the Pythagorean theorem on each right triangle:\r\n" );
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document.write( "  x² + y² = 5²\r\n" );
document.write( "  x² + y² = 25\r\n" );
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document.write( "So we have this system:\r\n" );
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document.write( "  x² + y² = 25\r\n" );
document.write( "      2xy = 24\r\n" );
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document.write( "Add equals to equals:\r\n" );
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document.write( "x² + 2xy + y² = 49\r\n" );
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document.write( "Factor the left side as a perfect square:\r\n" );
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document.write( "     (x + Y)² = 49\r\n" );
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document.write( "Use the principle of square roots\r\n" );
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document.write( "       x + y = ±7  but we can ignore the negative sign.\r\n" );
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document.write( "       x + y = 7\r\n" );
document.write( "           y = 7 - x\r\n" );
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document.write( "Substitute in 2xy = 24\r\n" );
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document.write( "           2xy = 24\r\n" );
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document.write( "       2x(7-x) = 24\r\n" );
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document.write( "Divide both sides by 2\r\n" );
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document.write( "        x(7-x) = 12\r\n" );
document.write( "         7x-x² = 12\r\n" );
document.write( "     -x²+7x-12 = 0\r\n" );
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document.write( "Multiply both sides by -1\r\n" );
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document.write( "      x²-7x+12 = 0\r\n" );
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document.write( "Factor the left side\r\n" );
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document.write( "    (x-4)(x-3) = 0\r\n" );
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document.write( "  x-4 = 0;  x-3 = 0\r\n" );
document.write( "    x = 4;    x = 3\r\n" );
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document.write( "Substitute each in y = 7 - x\r\n" );
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document.write( "    y = 3;    y = 4\r\n" );
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document.write( "That looks like two different solutions but actually\r\n" );
document.write( "they are both the same.  One diagonal is 2x = 2(4) = 8 cm\r\n" );
document.write( "The other diagonal is 2y = 2(3) = 6 cm.\r\n" );
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document.write( "Edwin

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