document.write( "Question 1116522: Find the area in square centimeter of the largest square that can be cut from a sector of a circle radius 8 cm and a central 120 degree? \n" ); document.write( "

Algebra.Com's Answer #731431 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "Let A and B be the points on circle O with radius 8 such that angle AOB is 120 degrees.

\n" ); document.write( "Let the square be CDEF, with C on radius OA and D on radius OB.

\n" ); document.write( "Let OG be the radius that bisects the square and the 120 sector of the circle, intersecting CD at H and EF at I.

\n" ); document.write( "Let the side of the square be 2x, so that CH=HD=FI=IE=x.

\n" ); document.write( "We need to find the area of the square, which is $\"%282x%29%5E2+=+4x%5E2\"$ .

\n" ); document.write( "In right triangle OHD, HD=x and angle D is 30 degrees, so $\"OH=x%2Fsqrt%283%29\"$ .

\n" ); document.write( "In right triangle OIE, I is the right angle and the legs are $\"IE=x\"$ and $\"OI=2x%2Bx%2Fsqrt%283%29\"$ . Then since OE is a radius with length 8,
\n" ); document.write( " $\"x%5E2+%2B+%282x%2Bx%2Fsqrt%283%29%29%5E2+=+8%5E2\"$
\n" ); document.write( " $\"x%5E2%2B4x%5E2%2B4x%5E2%2Fsqrt%283%29%2Bx%5E2%2F3+=+64\"$
\n" ); document.write( " $\"x%5E2%281%2B4%2B4%2Fsqrt%283%29%2B1%2F3%29+=+64\"$
\n" ); document.write( " $\"x%5E2%28%2816%2B4sqrt%283%29%29%2F3%29+=+64\"$
\n" ); document.write( " $\"x%5E2+=+64%2F%28%28%2816%2B4sqrt%283%29%29%2F3%29%29\"$
\n" ); document.write( "

\n" ); document.write( "Answer: The area of the square is $\"192%284-sqrt%283%29%29%2F13\"$ \n" ); document.write( "

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