Euclid theorems

Calculate the sides of a right triangle if leg a = 6 cm, and a section of the hypotenuse, which is located adjacent to the second leg b is 5cm.

Correct result:

a =  6 cm
b =  6.7082 cm
c =  9 cm

Solution:

$a=6=6\text{ cm}$

$$\begin{gathered} c_2=5\text{ cm } \\ h\mathrm{/}a=\sqrt{c_2^2-h^2}\mathrm{/}h \\ {h}^2\mathrm{/}a=\sqrt{c_2^2-h^2} \\ 5\sqrt{36-h^2}=h^2 \\ h=2\cdot \sqrt{5}=2\sqrt{5}\doteq 4.4721 \\ b=\sqrt{c_2^2+h^2}=\sqrt{5^2+4.4721^2}=3\sqrt{5}=6.7082\text{ cm} \end{gathered}$$

$$\begin{gathered} c=\sqrt{a^2+b^2}=\sqrt{6^2+6.7082^2}=9\text{ cm } \\ \text{ Verifying Solution: } \\ {c}_1=c-c_2=9-5=4\text{ cm } \\ {x}_1=a^2-c\cdot c_1=6^2-9\cdot 4=0 \\ {x}_2=b^2-c\cdot c_2=6.7082^2-9\cdot 5\doteq 8.4626\cdot 10^{-12} \end{gathered}$$

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