document.write( "Question 476259: In the diagram given below, three circles having centres at A, B and C touch each other externally.
\n" ); document.write( "(a) given that the radius of the three circles are 4cm, x cm, and (2x+2) cm respectively, and angle ABC=90°, form an equation in x and show that it reduces to 3x2-2x-8=0
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Algebra.Com's Answer #326632 by Edwin McCravy(20056) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( "AB = 4+x\r\n" );
document.write( "BC = x + (2x+2) = x + 2x+2 = 3x + 2\r\n" );
document.write( "AC = 4 + (2x+2) = 4 + 2x+2 = 2x + 6\r\n" );
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document.write( "By the Pythagorean theorem:\r\n" );
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document.write( "                 AB² + BC² = AC²\r\n" );
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document.write( "          (4+x)² + (3x+2)² = (2x+6)²\r\n" );
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document.write( " (4+x)(4+x) + (3x+2)(3x+2) = (2x+6)(2x+6)\r\n" );
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document.write( "  (16+8x+x²) + (9x²+12x+4) = 4x²+24x+36\r\n" );
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document.write( "        16+8x+x²+9x²+12x+4 = 4x²+24x+36\r\n" );
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document.write( "               10x²+20x+20 = 4x²+24x+36\r\n" );
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document.write( "                 6x²-4x-16 = 0\r\n" );
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document.write( "Divide through by 2\r\n" );
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document.write( "                  3x²-2x-8 = 0\r\n" );
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document.write( "That was the first thing you were asked to show.\r\n" );
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document.write( "Factor the left side:\r\n" );
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document.write( "               (3x+4)(x-2) = 0\r\n" );
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document.write( "            3x+4 = 0       x-2 = 0\r\n" );
document.write( "              3x = -4        x = 2\r\n" );
document.write( "               x = -4/3\r\n" );
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document.write( "We ignore the negative answer. \r\n" );
document.write( "So the value of x is 2.  That was the second \r\n" );
document.write( "thing you wanted. \r\n" );
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document.write( "We calculate the lengths of the sides:\r\n" );
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document.write( "AB = 4+x = 4+2 = 6\r\n" );
document.write( "BC = 3x+2 = 3(2)+2 = 6+2 = 8\r\n" );
document.write( "AC = 2x+6 = 2(2)+ 6 = 4+6 = 10\r\n" );
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document.write( "Now we want to find the length of the perpendicular from \r\n" );
document.write( "B to AC.  We'll draw that perpendicular AD in and label \r\n" );
document.write( "it h (in green):\r\n" );
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document.write( "Now the circles just get in the way, so we'll erase them\r\n" );
document.write( "and just look at the triangle, whose sides are\r\n" );
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document.write( "AB=6, BC=8, AC=10\r\n" );
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document.write( "So we can dispense with the circles and just keep the triangle and\r\n" );
document.write( "the green altitude AD = h:\r\n" );
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document.write( "A perpendicular from the right angle to the hypotenuse of a right \r\n" );
document.write( "triangle divides the right triangle into two smaller similar right\r\n" );
document.write( "triangles, both of which are similar to the whole right triangle.\r\n" );
document.write( "So\r\n" );
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document.write( "ᅀADB∼ᅀBDC∼ᅀABC\r\n" );
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document.write( "Using the fact that ᅀADB∼ᅀABC and the fact that AC=10\r\n" );
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document.write( "Cross-multiply:\r\n" );
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document.write( "So the length of the perpendicular AD is 4.8\r\n" );
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document.write( "Edwin

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