Ratio of squares

A circle is given, and a square is inscribed. The smaller square is inscribed in a circular arc formed by the square's side and the circle's arc. What is the ratio of the areas of the large and small squares?

Final Answer:

r =  5111486:48007

Step-by-step explanation:

$$\begin{gathered} r_1=1 \\ u=2r_1=\sqrt{2}{a}_1 \\ {a}_1=2\cdot r_1/\sqrt{2}=2\cdot 1/\sqrt{2}=\sqrt{2}\doteq 1.4142 \\ {r}_1^2=a_2^2+(2\cdot a_2+a_1/2) \\ {r}_1^2=x^2+(2\cdot x+a_1/2) \\ {1}^2=x^2+(2\cdot x+1.4142135623731/2) \\ -x^2-2x+0.293=0 \\ {x}^2+2x-0.293=0 \\ a=1;b=2;c=-0.293 \\ D=b^2-4ac=2^2-4\cdot 1\cdot (-0.293)=5.1715728753 \\ D>0 \\ {x}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{-2\pm \sqrt{5.17}}{2} \\ {x}_{1,2}=-1\pm 1.137055 \\ {x}_1=0.137054624 \\ {x}_2=-2.137054624 \\ {a}_2=x_1=0.1371\doteq 0.1371 \\ {r}_2=\sqrt{2}\cdot a_2/2=\sqrt{2}\cdot 0.1371/2\doteq 0.0969 \\ {S}_1=a_1^2=1.4142^2=2 \\ {S}_2=a_2^2=0.1371^2\doteq 0.0188 \\ r=S_1/S_2=2/0.0188\approx 106.4738=5111486:48007 \end{gathered}$$

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