Question

triunghiul ABC este dreptunghic in A. Daca AB/BC=1/2 si AB+AC=9 cm , calculati aria triunghiului ABC

Answer 1

AB/BC=1/2 =>BC=2AB
T.Pit. AB²+AC²=BC²
         AB²+AC²=4AB² =>AC²=3AB² sau AC=AB√3
AB+AC=9 =>AC=9-AB =>9-AB=AB√3 sau 9-AB-AB√3=0 l·(-1) =>AB+AB√3=9
AB(√3+1)=9 =>AB=9/(√3+1)=9(√3-1)/2
AC+ 9(√3-1)/2 = 9
2AC+9(√3-1)-18=0
2AC=18-9(√3-1)
2AC=18-9√3+9
2AC=27-9√3 =>AC=9(3-√3)/2
Aria unui triunghi dreptunghic este semisuma catetelor.
$A= \frac{ \frac{9( \sqrt{3}-1) }{2}* \frac{9(3- \sqrt{3)} }{2} }{2} = \frac{ \frac{81( \sqrt{3}-1)(3- \sqrt{3} ) }{4} }{2} = \frac{162(2 \sqrt{3}-3) }{8} = \frac{81(2 \sqrt{3}-3) }{8}$