# RT - hypotenuse and altitude

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

Result

gb =  36 m
gt =  81 m

#### Solution:

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

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