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 cm
c2=5 h/a=c22h2/h h2/a=c22h2 5 36h2=h2  h=2 5=2 54.4721 b=c22+h2=52+4.47212=3 5=6.708 cmc_{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=a2+b2=62+6.708228.9998 c1=cc2=8.999853.9998 x1=a2c c1=628.9998 3.99980.002 x2=b2c c2=6.708228.9998 50.002 c=8.9998=9 cmc=\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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