Question

Determinati raza cercului circumscris triunghiului dreptunghic cu lungimea unei catete egala cu 10 cm si raza cercului inscris egala cu 4cm

Answer 1

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Answer 2

[tex]\it Fie\ a, \ b,\ c - \ lungimile\ laturilor,\ c= 10. \\\;\\ \\\;\\ r = \dfrac{b+c-a}{2} \Rightarrow 4 =\dfrac{ b+10-a}{ 2} \Rightarrow 8 = 10+b-a \Rightarrow a = 10+b-8 \\\;\\ \\\;\\ \Rightarrow a = b+2 \ \ \ \ \ (*)[/tex]


Aplicăm teorema lui Pitagora :


$\it a^2=b^2+c^2 \stackrel{(*)}{\Longrightarrow} (b+2)^2 = b^2+10^2 \Longrightarrow (b+2)^2-b^2=100 \Longrightarrow \\\;\\ \Longrightarrow (b+2-b)(b+2+b) =100 \Longrightarrow 2(2b+2) = 100 \Longrightarrow \\\;\\ \Longrightarrow 4(b+1) = 100 |_{:4} \Longrightarrow b+1 = 25 \Longrightarrow b = 24$

[tex]\it a = b+2 \Longrightarrow a = 24+2 = 26 \\\;\\ R = \dfrac{a}{2} =\dfrac{26}{2} = 13[/tex]