Quatrefoil 81138

Gothic quatrefoil is an ornament in which four identical touching smaller circles are inscribed in a larger circle, as you can see in the picture. The radius of the great circle is one meter. Calculate the radius of the smaller circle in meters.

Correct answer:

r =  0.4142 m

Step-by-step explanation:

$$\begin{gathered} R=1\text{m} \\ (r+r{)}^2=(r+x{)}^2+(r+x{)}^2 \\ r+r+x=R \\ r=(R-x)/2 \\ x=R-2r \\ (2r{)}^2=2(r+x{)}^2 \\ (2r{)}^2=2(r+R-2r{)}^2 \\ 4r^2=2(R-r{)}^2 \\ 2r^2=(R-r{)}^2 \\ 2r^2=(1-r{)}^2 \\ {r}^2+2r-1=0 \\ a=1;b=2;c=-1 \\ D=b^2-4ac=2^2-4\cdot 1\cdot (-1)=8 \\ D>0 \\ {r}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{-2\pm \sqrt{8}}{2}=\frac{-2\pm 2\sqrt{2}}{2} \\ {r}_{1,2}=-1\pm 1.414214 \\ {r}_1=0.414213562 \\ {r}_2=-2.414213562 \\ r=r_1=0.4142\doteq 0.4142 \\ \text{ Verifying Solution: } \\ x=R-2\cdot r=1-2\cdot 0.4142\doteq 0.1716\text{m} \end{gathered}$$

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