document.write( "Question 641702: Starting at the center, designated as C, construct an isosceles right triangle with a congruent leg pointing north from the center, and the other congruent leg pointing east from the top of the first leg. The hypotenuse is at a 45-degree angle pointing northeast from C. Make each congruent leg 2 centimeters in length. Using the hypotenuse of the first triangle as one leg, and another 2-centimeter leg at right angles to it on the outside, construct a second right triangle. Again, the hypotenuse of this new triangle will be a segment coming from C and will point (approximately) east-by-northeast. Repeat, each time using the previous hypotenuse as the longer of the two legs of the new right triangle, and adding a leg of 2 centimeters to the outside end, then forming the new hypotenuse from C. Continue this process until your combined figure has wrapped all the way around the center, so the next triangle you construct overlaps the first one.
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Algebra.Com's Answer #403779 by Edwin McCravy(20055) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( "It has four names: Root Spiral, Root Snail, Spiral of Theodorus, and Wheel of Theodorus.\r\n" );
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document.write( "There are 16 triangles poosible before any overlapping occurs.\r\n" );
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document.write( "By using the Pythagorean theorem over and over, the first \r\n" );
document.write( "(shortest spoke, the vertical one) has length 2 cm and \r\n" );
document.write( "the last spoke (longest spoke) has length \r\n" );
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document.write( "The sequence of lengths of spokes radiating from center C going\r\n" );
document.write( "clockwise is:\r\n" );
document.write( "cm,cm, cm, cm, cm, cm, cm, cm, cm, cm, cm, cm, \r\n" );
document.write( "cm, cm, cm, cm,cm.\r\n" );
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document.write( "[This spiral is usually drawn with the first (vertical spoke being 1 unit\r\n" );
document.write( "instead of 2 cm. In other words, we could let 1 unit be 2 cm. and then those\r\n" );
document.write( "spoke lengths would be the square roots of the integers in units, not\r\n" );
document.write( "centimeters. They would be units,units, etc., \r\n" );
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document.write( "I'm sure the reason you were instructed to use 2 cm instead of 1 unit, was \r\n" );
document.write( "so that the area would be easier to calculate since there would be no\r\n" );
document.write( "fractions].\r\n" );
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document.write( "The area of each triangle is found by ×(first leg)×(second leg)\r\n" );
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document.write( "The length of the first leg of the 1st triangle is 2\r\n" );
document.write( "The length of the second leg of the 1st triangle is 2\r\n" );
document.write( "The area of the 1st triangle is ×2×2 = 2 cm²\r\n" );
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document.write( "Area of first triangle = 2cm²\r\n" );
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document.write( "The length of the first leg of the 2nd triangle is \r\n" );
document.write( "The length of the second leg of the 2nd triangle is 2\r\n" );
document.write( "The area of the 2nd triangle is ×2×2 = 2 cm²\r\n" );
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document.write( "Area of first two triangles = [2 + 2] cm²\r\n" );
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document.write( "The length of the first leg of the 3rd triangle is \r\n" );
document.write( "The length of the second leg of the 3rd triangle is 2\r\n" );
document.write( "The area of the 3rd triangle is ×2×2 = 2 cm²\r\n" );
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document.write( "Area of first three triangles = [2 + 2 + 2 ] cm²\r\n" );
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document.write( "So you can see that the area of the entire spiral is, after \r\n" );
document.write( "factoring out a 2:\r\n" );
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document.write( "A = \r\n" );
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document.write( "[Notice that the last spoke,  is not needed since it is\r\n" );
document.write( "not used to find the area of the 16th triangle]\r\n" );
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document.write( "I get that total area to be 88.9383932 cm². Better check me on that. \r\n" );
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document.write( "Edwin

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