# 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?

Result

a =  24 m
b =  18 m

#### Solution:

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

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

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