document.write( "Question 1043104: Two perpendicular chords divide a circle with a radius of 13 cm into four parts. If the perpendicular distances of both chords are 5 cm each from the center of the circle, find the area of the smallest part. \n" ); document.write( "

Algebra.Com's Answer #658261 by ikleyn(52782) $\"\"$ $\"About$

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\n" ); document.write( "Two perpendicular chords divide a circle with a radius of 13 cm into four parts.
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document.write( "The Figure is on the right.\r\n" );
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document.write( "AB and CD are two given perpendicular chords.\r\n" );
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document.write( "Let the Point E be their intersection point.\r\n" );
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document.write( "We need to find the area of the shape EBC.\r\n" );
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document.write( "Draw the radius OF through the point E.\r\n" );
document.write( "It is clear that this radius cuts the shape EBC in two congruent parts,           \r\n" );
document.write( "EBF and EFC, and each of them has the area half of the area EBC.\r\n" );
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document.write( "Also draw the radius OX (horizontal line) and the radius OC.\r\n" );
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document.write( "Let G is the intersection of OX and CD.\r\n" );
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document.write( "Then  |OG] = 5,  |OC| = 13, and the triangle OGC is a right-angled.\r\n" );
document.write( "Then  |GC| = 12  (Pythagorean triangle 5, 12, 13).\r\n" );
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document.write( "The angle COX =  = 1.176 radians.  \r\n" );
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document.write( "The angle FOX = 45° =  radians. \r\n" );
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document.write( "          Figure.\r\n" );
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Then the angle COF =  =  = 0.391 radians, and we can consider it as a known value.\r\n" );
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document.write( "Let us denote this angle COF as :  = 0.391 radians.\r\n" );
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document.write( "Now we are on the straight finish line.\r\n" );
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document.write( "The area of the sector COF is  = 33.04 , and all we need to do is to distract the area of the triangle OEC.\r\n" );
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document.write( "The area of the triangle OEC is .|EF|*|OG| =  = 17.5 .   (the segments EF and OG are the base and the altitude of the triangle)\r\n" );
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document.write( "Hence, the shape EFC has the area  33.04 - 17.5 = 15.54 .\r\n" );
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document.write( "Then the shape EBC has the area twice of it: 2*15.54 = 37.08 .\r\n" );
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document.write( "Answer.  The shape  EBC  has the area of 37.08 .\r\n" );
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