Semicircles

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