Land boundary

The land has the shape of a right triangle. The hypotenuse has a length of 30m. The circumference of the land is 72 meters. What is the length of the remaining sides of the land boundary?

Correct result:

a =  24 m
b =  18 m

Solution:

$$\begin{gathered} c=30\text{ m } \\ o=72\text{ m } \\ {c}^2=a^2+b^2 \\ o=a+b+c \\ {c}^2=a^2+(o-a-c{)}^2 \\ 30^2=a^2+(42-a{)}^2 \\ 30^2=a^2+(42-a{)}^2 \\ -2a^2+84a-864=0 \\ 2a^2-84a+864=0 \\ p=2;q=-84;r=864 \\ D=q^2-4pr=84^2-4\cdot 2\cdot 864=144 \\ D>0 \\ {a}_{1,2}={\displaystyle \frac{-q\pm \sqrt{D}}{2p}}={\displaystyle \frac{84\pm \sqrt{144}}{4}} \\ {a}_{1,2}={\displaystyle \frac{84\pm 12}{4}} \\ {a}_{1,2}=21\pm 3 \\ {a}_1=24 \\ {a}_2=18 \\ \text{ Factored form of the equation: } \\ 2(a-24)(a-18)=0 \\ a=a_1=24=24\text{ m} \end{gathered}$$

Our quadratic equation calculator calculates it.

$$\begin{gathered} b=a_2 \\ b=o-a-c=72-24-30=18\text{ m} \end{gathered}$$

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