Without Euclid laws

Right triangle ABC with right angle at the C has a=14 and hypotenuse c=26. Calculate the height h of this triangle without the use of Euclidean laws.

Correct answer:

h =  11.8

Step-by-step explanation:

$$\begin{gathered} a=14 \\ c=26 \\ {c}^2=a^2+b^2 \\ b=\sqrt{c^2-a^2}=\sqrt{26^2-14^2}=4\sqrt{30}\doteq 21.9089 \\ {S}_1=S_2 \\ {\displaystyle \frac{ab}{2}}={\displaystyle \frac{ch}{2}} \\ a\cdot b=c\cdot h \\ h={\displaystyle \frac{a\cdot b}{c}}={\displaystyle \frac{14\cdot 21.9089}{26}}=11.8 \end{gathered}$$

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