Five circles

On the line segment CD = 6 there are 5 circles with a radius one at regular intervals. Find the lengths of the lines AD, AF, AG, BD, and CE.

Correct answer:

d =  4.5826
f =  2.1794
g =  3.1225
b =  3.6056
c =  2.6458

Step-by-step explanation:

$$\begin{gathered} r=1 \\ h=v(A,CD) \\ A{S}_1{S}_2=\mathrm{\Delta }1-1-1,\alpha =\beta =\gamma =60\mathrm{°}, \\ h=\sqrt{3}\mathrm{/}2\cdot r=\sqrt{3}\mathrm{/}2\cdot 1\doteq 0.866 \\ {x}_1=(4+1\mathrm{/}2)\cdot r=(4+1\mathrm{/}2)\cdot 1={\displaystyle \frac{9}{2}}=4\frac{1}{2}=4.5 \\ d=\mathrm{\mid }AD\mathrm{\mid } \\ d=\sqrt{x_1^2+h^2}=\sqrt{4.5^2+0.866^2}=\sqrt{21}=4.5826 \end{gathered}$$

$$\begin{gathered} f=\mathrm{\mid }AE\mathrm{\mid } \\ {x}_2=4\mathrm{/}2\cdot r=4\mathrm{/}2\cdot 1=2 \\ f=\sqrt{x_2^2+h^2}=\sqrt{2^2+0.866^2}=2.1794 \end{gathered}$$

$$\begin{gathered} g=\mathrm{\mid }AG\mathrm{\mid } \\ {x}_3=6\mathrm{/}2\cdot r=6\mathrm{/}2\cdot 1=3 \\ g=\sqrt{x_3^2+h^2}=\sqrt{3^2+0.866^2}=3.1225 \end{gathered}$$

$$\begin{gathered} b=\mathrm{\mid }BD\mathrm{\mid } \\ {x}_4=7\mathrm{/}2\cdot r=7\mathrm{/}2\cdot 1={\displaystyle \frac{7}{2}}=3\frac{1}{2}=3.5 \\ b=\sqrt{x_4^2+h^2}=\sqrt{3.5^2+0.866^2}=\sqrt{13}=3.6056 \end{gathered}$$

$$\begin{gathered} c=\mathrm{\mid }CE\mathrm{\mid } \\ {x}_5=5\mathrm{/}2\cdot r=5\mathrm{/}2\cdot 1={\displaystyle \frac{5}{2}}=2\frac{1}{2}=2.5 \\ c=\sqrt{x_5^2+h^2}=\sqrt{2.5^2+0.866^2}=\sqrt{7}=2.6458 \end{gathered}$$

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