# 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:

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

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$b=a_{2} \ \\ \ \\ b=o-a-c=72-24-30=18 \ \text{m}$

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