Question

Sa se determine lungimile catetelor unui triunghi dreptunghic daca suma lor este 17, iar aria este 30.

Answer 1

Răspuns:

Explicație pas cu pas:

Answer 2

Fie a, b lungimile catetelor.

$\it a+b=17 \Rightarrow b=17-a\ \ \ \ \ (*)\\ \\ \mathcal{A}=\dfrac{a\cdot b}{2}=30|_{\cdot2} \Rightarrow a\cdot b=60\ \stackrel{(*)}{\Longrightarrow}\ a(17-a)=60 \Rightarrow 17a-a^2=60 \Rightarrow \\ \\ \Rightarrow 0=a^2-17a+60 \Rightarrow a^2-17a+60=0 \Rightarrow a^2-5a-12a+60=0 \Rightarrow \\ \\ \Rightarrow a(a-5)-12(a-5)=0 \Rightarrow (a-5)(a-12)=0 \Rightarrow a_1=5,\ \ a_2=12$

$\it a=5\ \stackrel{(*)}{\Longrightarrow}\ b=12\\ \\ a=12\ \stackrel{(*)}{\Longrightarrow}\ b=5\\ \\ Deci:\ \ (a,\ b) \in\{(5,\ 12),\ \ (12,\ 5)\}$