Hypotenuse and height

In a right triangle is length of the hypotenuse c = 56 cm and height h_c = 4 cm. Determine the length of both trangle legs.

Correct result:

a =  55.9 cm
b =  4 cm

Solution:

$$\begin{gathered} 56=x+y \\ {4}^2=xy \\ x(56-x)=4^2 \\ {x}^2-56x+16=0 \\ a=1;b=-56;c=16 \\ D=b^2-4ac=56^2-4\cdot 1\cdot 16=3072 \\ D>0 \\ {x}_{1,2}={\displaystyle \frac{-b\pm \sqrt{D}}{2a}}={\displaystyle \frac{56\pm \sqrt{3072}}{2}}={\displaystyle \frac{56\pm 32\sqrt{3}}{2}} \\ {x}_{1,2}=28\pm 27.7128129211 \\ {x}_1=55.7128129211 \\ {x}_2=0.287187078898 \\ \text{ Factored form of the equation: } \\ (x-55.7128129211)(x-0.287187078898)=0 \\ a=\sqrt{x_1\cdot 56}=55.9\text{ cm} \end{gathered}$$

$b=\sqrt{x_2\cdot 56}=4\text{ cm}$

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