Question

Raportul dintre catetele unui triunghi dreptunghic este 4 supra 3,iar ipotenuza este de 80 cm.Aflati catetele si inaltimea triunghiului dreptunghic

Answer 1

   
[tex]\displaystyle \\ \texttt{Se da: } \\ \Delta ABC ~~cu~~\ \textless \ A=90^o \\ BC = 80 cm \\ \\ \frac{AB}{AC} = \frac{4}{3} \\ \\ AD \perp BC ~~~D \in BC~~~(inaltime)\\ \texttt{Se cere: } \\ AB=? \\ AC=? \\ AD=? \\ \\ Rezolvare: \\ \\ \frac{AB}{AC} = \frac{4}{3} \\ \\ \Longrightarrow~~AB =4k~~si~~ AC=3k \\ \texttt{Aplicam Pitagora:} \\ AB^2+AC^2 = BC^2 \\ (4k)^2 + (3k)^2=80^2 \\ 16k^2 + 9K^2 = 6400 \\ k^2(16+9) = 6400 \\ 25 k^2 = 1600 \\ \\ k^2 = \frac{6400}{25}[/tex]


[tex]\displaystyle \\ k = \sqrt{\frac{6400}{25}} =\frac{80}{5} = \boxed{16} \\ \\ AB = 4k = 4\times 16 = \boxed{64 ~cm} \\ ;AC = 3k = 3\times 16 = \boxed{48 ~cm} \\ \\ \texttt{Calculam inaltimea:} \\ BC \times AD =AB \times AC\\ \\ AD = \frac{AB \times AC}{BC}= \frac{64 \times 48}{80}= \frac{8 \times 48}{10}=\frac{384}{10}= \boxed{38,4 ~cm} [/tex]