document.write( "Question 1066301: AB is the diameter of a circle. AD and BC are tangents to the circle with AD = 9cm and BC = 16cm. If AC and BD intersect at a point on the circle, then the length, in centimetres, of AB is:
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Algebra.Com's Answer #681716 by ikleyn(52787) $\"\"$ $\"About$

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\n" ); document.write( "AB is the diameter of a circle. AD and BC are tangents to the circle with AD = 9cm and BC = 16cm.
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document.write( "0.  Make a sketch.\r\n" );
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document.write( "    Let \"x\" be the length of AB, which is under the question: x = |AB|.\r\n" );
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document.write( "1.  Let P be the intersection point of the segments AC and BD. \r\n" );
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document.write( "    As the condition says, the point P lies on the circle.\r\n" );
document.write( "    Therefore, the angle APB is the right angle (it leans on the diameter AB !)\r\n" );
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document.write( "    Thus, the segments AC and BD are perpendicular.\r\n" );
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document.write( "    Also, let's denote a = |BP|, b = |AP|.\r\n" );
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document.write( "2.  Right-angled triangles ABC and APB are similar (they have the common acute angle BAP).\r\n" );
document.write( "    Therefore, their corresponding sides are proportional:  = ,  or   = .    (1)\r\n" );
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document.write( "3.  Right-angled triangles BAD and BPA are similar (they have the common acute angle ABP).\r\n" );
document.write( "    Therefore, their corresponding sides are proportional:  = ,  or   = .    (2)\r\n" );
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document.write( "4.  Divide (1) by (2) (both sides). You will get  = , which implies  =  = .\r\n" );
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document.write( "5.  Now substitute  instead of  into (1).\r\n" );
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document.write( "    You will get  = , which implies x =  = 12.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Answer. |AB| = 12 cm. Option c).\r
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