11990 perimeter RT

A right triangle has integer side lengths and a perimeter of 11990. In addition, we know that one of its perpendiculars has a prime number length. Find its length.

Correct answer:

a =  109

Step-by-step explanation:

$$\begin{gathered} o=11990 \\ o=a+b+c=11990=2\times 5\times 11\times 109 \\ a\cdot a+b\cdot b=c\cdot c \\ a\cdot a+b\cdot b=(11990-a-b{)}^2 \\ {a}^2+b^2=143760100-23980a+a^2-23980b+2ab+b^2 \\ 23980a+23980b=143760100+2ab \\ a=109 \\ 23980\cdot a+23980\cdot b=143760100+2\cdot a\cdot b \\ 23980\cdot 109+23980\cdot b=143760100+2\cdot 109\cdot b \\ 23762b=141146280 \\ b=\frac{141146280}{23762}=5940 \\ b=5940 \\ c=o-a-b=11990-109-5940=5941 \end{gathered}$$

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