Semicircles

In a rectangle with sides of 4 cm and 8 cm, there are two different semicircles, each with its endpoints at adjacent vertices and touching the opposite side.
Construct a square such that two of its vertices lie on one semicircle, the other two vertices lie on the other semicircle, and its sides are parallel to the sides of the rectangle.

Final Answer:

a =  7.2915 cm

Step-by-step explanation:

$$\begin{gathered} x=4\text{cm} \\ y=8\text{cm} \\ {a}^2-4a-24=0 \\ {a}^2-4a-24=0 \\ p=1;q=-4;r=-24 \\ D=q^2-4pr=4^2-4\cdot 1\cdot (-24)=112 \\ D>0 \\ {a}_{1,2}=\frac{-q\pm \sqrt{D}}{2p}=\frac{4\pm \sqrt{112}}{2}=\frac{4\pm 4\sqrt{7}}{2} \\ {a}_{1,2}=2\pm 5.291503 \\ {a}_1=7.291502622 \\ {a}_2=-3.291502622 \\ a=a_1=7.2915=7.2915\text{cm} \end{gathered}$$

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