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Two squares in a circle (Posted on 2014-12-05) Difficulty: 3 of 5

I drew a square of side 5 together with its circumcircle. At the middle of the arc between a side and the circle, I drew a smaller square with two vertices on the side, and the other two on the circle.

What was the area of the smaller square?

See The Solution Submitted by broll    
Rating: 3.0000 (2 votes)

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Solution Solution Comment 4 of 4 |
As the diagonal of the larger square is congruent to the diameter of the circle, it can be calculated as 5√2, thus the radius of the circle is (5/2)√2.
Let the length of a side of the smaller square be a. Then, as half the length of the bigger square is 5/2, one vertex of the smaller square will will touch the circle at point (a/2, (5/2+a)). Substituting this point in the equation of a circle: 
(a/2)2 + ((5+2a)/2)2 = ((5/2)√2)2.
Simplifying and arranging the quadratic equation, it becomes:
5a2 + 20a – 25 = 0
Solving for the roots of the quadratic: a1 1 = 1, a2 = -5
As the length is positive, the first root, a1, is the length of the side of the square, i.e., 1.  
  Posted by Dej Mar on 2014-12-06 06:46:01
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