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# geometry

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A circle is inscribed in a square. A smaller circle is drawn tangent to two sides of the square and externally tangent to the inscribed circle. Find the area of the blue shaded region to two decimal places. Dec 25, 2020

#1
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I don't know how to solve for the smaller circle, but maybe we can first find the area of the blue region without the circle, that would be the area of the square - the circle, so the area of the square is (5*2)^2, since the radius is 5, the Diameter is 10 which is the side of the square, 100 is the area. then the circle is (pi)(r)^2, the radius is 5 so (pi)(25), 100-25pi, then divided by 4 because 4 regions is 25-25/4(pi), sorry idk how to solve the area of smaller circle but that's the idea, maybe Cphill or somebody wil give a splendid answer.

Dec 25, 2020
#2
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We can construct a rifght  triangle with legs of  (5 - r)   and a hyotenuse of  (5 + r)  where r is the radius of the smaller circle

So by the Pythagorean Theorem  we have

2 ( 5 - r)^2  =  (5 + r)^2     simplify

2 ( r^2 - 10r + 25)  =  r^2  + 10r   + 25

2r^2 - 20r + 50   =r^2  + 10r   + 25

r^2 - 30r  +  25  =  0       complete the square on r

r^2 - 30r  + 225  =  -25 + 225

(r - 15)^2   = 200       take the smaller root

r  - 15  =  -sqrt (200)

r =   15  - 10sqrt (2)

So  the blue area   =    (area of quarter square    -area of  quarter circle  -area of small circle ) =

10^2/4  - (25/4)pi   - (15 - 10sqrt (2))^2*pi   ≈  3.05 units^2   Dec 25, 2020
edited by CPhill  Dec 25, 2020
#3
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DX / DY = DY / DZ

DX = DY2 / DZ        Everything else is easy.  Dec 26, 2020