Free space in the garden

The grandfather's free space in the garden was in the shape of a rectangular triangle with 5 meters and 12 meters in length. He decided to divide it into two parts and the height of the hypotenuse. For the smaller part creates a rock garden, for the larger sows grass. How many square meters has a larger part?

Correct result:

S =  25.5621 m²

Solution:

$$\begin{gathered} a=5\text{ m } \\ b=12\text{ m } \\ c=\sqrt{a^2+b^2}=\sqrt{5^2+12^2}=13 \\ {S}_1={\displaystyle \frac{a\cdot b}{2}}={\displaystyle \frac{5\cdot 12}{2}}=30{\text{m}}^2 \\ {S}_1={\displaystyle \frac{c\cdot v}{2}} \\ v={\displaystyle \frac{2\cdot S_1}{c}}={\displaystyle \frac{2\cdot 30}{13}}={\displaystyle \frac{60}{13}}\doteq 4.6154\text{ m } \\ c{c}_1=a^2 \\ c{c}_2=b^2 \\ {c}_1={\displaystyle \frac{a^2}{c}}={\displaystyle \frac{5^2}{13}}={\displaystyle \frac{25}{13}}\doteq 1.9231\text{ m } \\ {c}_2={\displaystyle \frac{b^2}{c}}={\displaystyle \frac{12^2}{13}}={\displaystyle \frac{144}{13}}\doteq 11.0769\text{ m } \\ S={\displaystyle \frac{c_2\cdot v}{2}}={\displaystyle \frac{11.0769\cdot 4.6154}{2}}={\displaystyle \frac{4320}{169}}=25.5621{\text{m}}^2 \end{gathered}$$

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