# RT sides

Find the sides of a rectangular triangle if legs a + b = 17cm and the radius of the written circle ρ = 2cm.

Correct result:

c =  13 cm
a =  12 cm
b =  5 cm

#### Solution:

$a+b=17 \ \\ r=2 \ \text{cm} \ \\ \ \\ r=\dfrac{ a+b-c }{ 2 } \ \\ \ \\ a+b=c+2r \ \\ \ \\ c=17 - 2 \cdot \ r=17 - 2 \cdot \ 2=13 \ \text{cm}$
$a^2+b^2=c^2 \ \\ a^2 + (17-a)^2=13^2 \ \\ \ \\ 2a^2 -34a +120=0 \ \\ \ \\ p=2; q=-34; r=120 \ \\ D=q^2 - 4pr=34^2 - 4\cdot 2 \cdot 120=196 \ \\ D>0 \ \\ \ \\ a_{1,2}=\dfrac{ -q \pm \sqrt{ D } }{ 2p }=\dfrac{ 34 \pm \sqrt{ 196 } }{ 4 } \ \\ a_{1,2}=\dfrac{ 34 \pm 14 }{ 4 } \ \\ a_{1,2}=8.5 \pm 3.5 \ \\ a_{1}=12 \ \\ a_{2}=5 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ 2 (a -12) (a -5)=0 \ \\ \ \\ a=a_{1}=12 \ \text{cm}$

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$b=17-a=17-12=5 \ \text{cm}$

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