Without Euclid laws

Right triangle ABC with right angle at the C has a=5 and hypotenuse c=19. Calculate the height h of this triangle without the use of Euclidean laws.

Result

h =  4.82

Solution:

b2=c2−a2 b=192−52=18.33  S1=S2 ab2=ch2 ab=ch  h=abc=5⋅18.3319=4.82b^2 = c^2 - a^2 \ \\ b = \sqrt{ 19^2 - 5^2} = 18.33 \ \\ \ \\ S_1 = S_2 \ \\ \dfrac{ ab}{2} = \dfrac{ c h }{2} \ \\ ab = c h \ \\ \ \\ h =\dfrac{ ab }{ c} = \dfrac{ 5 \cdot 18.33 }{ 19 } = 4.82

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Pythagorean theorem is the base for the right triangle calculator.
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