RT sides

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

Correct answer:

c =  13 cm
a =  12 cm
b =  5 cm

Step-by-step explanation:

$$\begin{gathered} a+b=17\text{ cm } \\ r=2\text{ cm } \\ r={\displaystyle \frac{a+b-c}{2}} \\ a+b=c+2r \\ c=17-2\cdot r=17-2\cdot 2=13\text{ cm} \end{gathered}$$

$$\begin{gathered} a^2+b^2=c^2 \\ {a}^2+(17-a{)}^2=13^2 \\ 2a^2-34a+120=0 \\ p=2;q=-34;r=120 \\ D=q^2-4pr=34^2-4\cdot 2\cdot 120=196 \\ D>0 \\ {a}_{1,2}={\displaystyle \frac{-q\pm \sqrt{D}}{2p}}={\displaystyle \frac{34\pm \sqrt{196}}{4}} \\ {a}_{1,2}={\displaystyle \frac{34\pm 14}{4}} \\ {a}_{1,2}=8.5\pm 3.5 \\ {a}_1=12 \\ {a}_2=5 \\ \text{ Factored form of the equation: } \\ 2(a-12)(a-5)=0 \\ a=a_1=12\text{ cm} \end{gathered}$$

Our quadratic equation calculator calculates it.

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

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