RT - hypotenuse and altitude

Right triangle BTG has hypotenuse g=117 m and altitude to g is 54 m.
How long are hypotenuse segments?

Correct result:

g_b =  36 m
g_t =  81 m

Solution:

$$\begin{gathered} g_b+g_t=117 \\ {g}_b\cdot g_t=54^2 \\ {g}^2-117g+2916=0 \\ a=1;b=-117;c=2916 \\ D=b^2-4ac=117^2-4\cdot 1\cdot 2916=2025 \\ D>0 \\ {g}_{1,2}={\displaystyle \frac{-b\pm \sqrt{D}}{2a}}={\displaystyle \frac{117\pm \sqrt{2025}}{2}} \\ {g}_{1,2}={\displaystyle \frac{117\pm 45}{2}} \\ {g}_{1,2}=58.5\pm 22.5 \\ {g}_1=81 \\ {g}_2=36 \\ \text{ Factored form of the equation: } \\ (g-81)(g-36)=0 \\ {g}_b=36\text{ m} \end{gathered}$$

$g_t=81\text{ m}$

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