Medians in right triangle

It is given a right triangle, angle C is 90 degrees. I know it medians t1 = 8 cm and median t2 = 12 cm. .. How to calculate the length of the sides?

Correct answer:

a =  11.6847 cm
b =  5.465 cm
c =  12.8996 cm

Step-by-step explanation:

$$\begin{gathered} t_1=8\text{ cm } \\ {t}_2=12\text{ cm } \\ {t}_1^2=x^2+(2y{)}^2 \\ {t}_2^2=y^2+(2x{)}^2 \\ {x}^2=t_1^2-4y^2 \\ y=\sqrt{{\displaystyle \frac{4\cdot t_1^2-t_2^2}{15}}}=\sqrt{{\displaystyle \frac{4\cdot 8^2-12^2}{15}}}\doteq 2.7325\text{ cm } \\ x=\sqrt{t_1^2-4\cdot y^2}=\sqrt{8^2-4\cdot 2.7325^2}\doteq 5.8424\text{ cm } \\ a=2\cdot x=2\cdot 5.8424=11.6847\text{ cm} \end{gathered}$$

$b=2\cdot y=2\cdot 2.7325=5.465\text{ cm}$

$$\begin{gathered} a^2+b^2=c^2 \\ c=\sqrt{a^2+b^2}=\sqrt{11.6847^2+5.465^2}=12.8996\text{ cm} \end{gathered}$$

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