Question

calculati lungimea laturilor unui triunghi dreptunghic cu catetele de x+1 si x si ipotmuza de 2x

Answer 1

Metoda 1

$(x+1)^2+x^2=(2x)^2 \\ x^2+2x+1+x^2=4x^2 \\ 2x+1=2x^2 \\ 2x^2-2x-1=0 \\ x_1= \frac{1+ \sqrt{3} }{2}, x_2= \frac{1-\sqrt{3}}{2}<0 \\ AB= \frac{1+ \sqrt{3} }{2}, AC= \frac{3+\sqrt{3} }{2}, BC=1+\sqrt{3}$

Metoda 2

$sinC= \frac{AB}{BC}= \frac{x}{2x}= \frac{1}{2} \\ C=30^o \\ cos C= \frac{AC}{BC}= cos30^o= \frac{ \sqrt{3} }{2} = \frac{x+1}{2x} \\ x \sqrt{3}=x+1 \\ x(\sqrt{3}-1)=1 \\ x= \frac{1}{\sqrt{3}-1}= \frac{\sqrt{3}+1}{2} =AB \\ AC= \frac{3+\sqrt{3}}{2}, BC=\sqrt{3}+1$