document.write( "Question 1060632: In the figure, ACB is an arc of a circle, and CD is the perpendicular bisector of chord AB. If CD =18 and AB = 12, find the area of the entire circle.
\n" ); document.write( "Image: http://prntscr.com/dgqi9q \n" ); document.write( "

Algebra.Com's Answer #675537 by ikleyn(52790) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( " In the figure, ACB is an arc of a circle, and CD is the perpendicular bisector of chord AB. If CD =18 and AB = 12, find the area of the entire circle.
\n" ); document.write( "Image: http://prntscr.com/dgqi9q
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "First, notice that perpendicular bisector of EVERY chord in a circle passes through the center of the circle.\r\n" );
document.write( "\r\n" );
document.write( "     This statement is easy to prove and it belongs to the area of \"common knowledge\", so I will not prove it.\r\n" );
document.write( "\r\n" );
document.write( "In particular, our perpendicular bisector CD passes through the center of the circle.\r\n" );
document.write( "Let O be the center of the circle. Then CD passes through O.  So, CD is the part of the diameter of the circle.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Second, let us proceed the segment CD till the intersection with the circle at the point E.\r\n" );
document.write( "It is not visible in the Figure, but I recommend you to make your own sketch to follow my arguments.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then the well known property of chords intersecting inside a circle says that\r\n" );
document.write( "\r\n" );
document.write( "|CD|*|DE| = |BD|*|DA|,  or 18*|DE|  = 6*6.   ( here 6 =  )\r\n" );
document.write( "\r\n" );
document.write( "Hence,  |DE| =  = 2.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then the get the remarkable fact: the diameter of the circle is  |CD| + |DE| = 18 + 2 = 20 units.\r\n" );
document.write( "\r\n" );
document.write( "Hence, the radius of the circle is 10 units.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Can you complete the solution from this point on your own?\r\n" );
document.write( "

\n" ); document.write( " \n" ); document.write( "

\n" );