# Euclid theorems

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

Correct result:

a =  6 cm
b =  6.708 cm
c =  9 cm

#### Solution:

$a=6 \ \text{cm}$
$c_{2}=5 \ \\ h/a=\sqrt{ c_{2}^2-h^2 }/h \ \\ h^2/a=\sqrt{ c_{2}^2-h^2 } \ \\ 5 \ \sqrt{ 36-h^2 }=h^2 \ \\ \ \\ h=2 \cdot \ \sqrt{ 5 }=2 \ \sqrt{ 5 } \doteq 4.4721 \ \\ b=\sqrt{ c_{2}^2+h^2 }=\sqrt{ 5^2+4.4721^2 }=3 \ \sqrt{ 5 }=6.708 \ \text{cm}$
$c=\sqrt{ a^2+b^2 }=\sqrt{ 6^2+6.7082^2 } \doteq 8.9998 \ \\ c_{1}=c-c_{2}=8.9998-5 \doteq 3.9998 \ \\ x_{1}=a^2-c \cdot \ c_{1}=6^2-8.9998 \cdot \ 3.9998 \doteq 0.002 \ \\ x_{2}=b^2-c \cdot \ c_{2}=6.7082^2-8.9998 \cdot \ 5 \doteq -0.002 \ \\ c=8.9998=9 \ \text{cm}$

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