Question

Perimetrul unui triunghi dreptunghic isoscel este de 12 cm. Aflati aria.

Answer 1

P=I+2c   i²=2c² i=  c√2⇒c√2+2c=12  ⇔c(√2+2)=12  ⇒c=12/(2+√2)
A=c²/2=144(2+√2)²/2=72(4+4√2+2)=72(6+4√2) cm²

Answer 2

       
$\text{Notam catetele cu x} \\ \text{Aplicam Pitagora si calculam ipotenuza} \\ Ipotenuza = \sqrt{ x^{2} + x^{2} } =\sqrt{ 2x^{2} } =x \sqrt{2} \\ P=x+x+x \sqrt{2}=12 \\ 2x+x \sqrt{2}=12 \\ x(2+ \sqrt{2}) =12 \\ x= \frac{12}{2+ \sqrt{2}} =\frac{12(2- \sqrt{2})}{(2+ \sqrt{2})(2- \sqrt{2})} =\frac{12(2- \sqrt{2})}{4- 2}=\frac{12(2- \sqrt{2})}{2}=6(2- \sqrt{2}) \\ => x = \boxed{6(2- \sqrt{2})}$

$Aria = \frac{ x^{2} }{2}=\frac{ (6(2- \sqrt{2}))^{2} }{2} =\frac{ 36(2- \sqrt{2})^{2}) }{2}=18(4-4 \sqrt{2}+2 )= \\ =18(6-4\sqrt{2})=\boxed{36(3-2\sqrt{2}) \;cn^2}$