SOLUTION: 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?

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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?
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Let A and B be the points on circle O with radius 8 such that angle AOB is 120 degrees.

Let the square be CDEF, with C on radius OA and D on radius OB.

Let OG be the radius that bisects the square and the 120 sector of the circle, intersecting CD at H and EF at I.

Let the side of the square be 2x, so that CH=HD=FI=IE=x.

We need to find the area of the square, which is %282x%29%5E2+=+4x%5E2.

In right triangle OHD, HD=x and angle D is 30 degrees, so OH=x%2Fsqrt%283%29.

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,
x%5E2+%2B+%282x%2Bx%2Fsqrt%283%29%29%5E2+=+8%5E2
x%5E2%2B4x%5E2%2B4x%5E2%2Fsqrt%283%29%2Bx%5E2%2F3+=+64
x%5E2%281%2B4%2B4%2Fsqrt%283%29%2B1%2F3%29+=+64
x%5E2%28%2816%2B4sqrt%283%29%29%2F3%29+=+64
x%5E2+=+64%2F%28%28%2816%2B4sqrt%283%29%29%2F3%29%29


Answer: The area of the square is 192%284-sqrt%283%29%29%2F13