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?

Result

S =  25.562 m²

Solution:

$a=5 \ \text{m} \ \\ b=12 \ \text{m} \ \\ \ \\ c=\sqrt{ a^2+b^2 }=\sqrt{ 5^2+12^2 }=13 \ \\ \ \\ S_{1}=\dfrac{ a \cdot \ b }{ 2 }=\dfrac{ 5 \cdot \ 12 }{ 2 }=30 \ \text{m}^2 \ \\ S_{1}=\dfrac{ c \cdot \ v }{ 2 } \ \\ \ \\ v=\dfrac{ 2 \cdot \ S_{1} }{ c }=\dfrac{ 2 \cdot \ 30 }{ 13 } \doteq \dfrac{ 60 }{ 13 } \doteq 4.6154 \ \text{m} \ \\ \ \\ c c_{1}=a^2 \ \\ c c_{2}=b^2 \ \\ \ \\ c_{1}=\dfrac{ a^2 }{ c }=\dfrac{ 5^2 }{ 13 } \doteq \dfrac{ 25 }{ 13 } \doteq 1.9231 \ \text{m} \ \\ c_{2}=\dfrac{ b^2 }{ c }=\dfrac{ 12^2 }{ 13 } \doteq \dfrac{ 144 }{ 13 } \doteq 11.0769 \ \text{m} \ \\ \ \\ S=\dfrac{ c_{2} \cdot \ v }{ 2 }=\dfrac{ 11.0769 \cdot \ 4.6154 }{ 2 } \doteq \dfrac{ 4320 }{ 169 } \doteq 25.5621 \doteq 25.562 \ \text{m}^2$

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