Atunci cand problema iti cere aria triunghiului si ti se dau toate laturile, poti sa:
Formula lui Herron:
[tex] \boxed{A=\sqrt{p(p-a)(p-b)(p-c)}}[/tex]
1. Verficam daca triunghiul e dreptunghic
18 > 9√3 > 8√3
18²=324
(9√3)²+(8√3)² = 81·3+64·3 = 243 + 192 = 435
Deci 18²≠(9√3)²+(8√3)² iar triunghiul nu e dreptunghic
OBS: Triunghiul, evident, nu este echilateral deoarece nu are toate laturile egale.
p=(18+9√3+8√3)/2=(18+17√3)/2
[tex]A=\sqrt{\frac{18+17\sqrt{3}}{2}(\frac{18+17\sqrt{3}}{2}-9\sqrt{3})(\frac{18+17\sqrt{3}}{2}-8\sqrt{3})(\frac{18+17\sqrt{3}}{2}-18)}[/tex]
[tex]A=\sqrt{\frac{18+17\sqrt{3}}{2}(\frac{18+17\sqrt{3}}{2}-\frac{18\sqrt{3}}{2})(\frac{18+17\sqrt{3}}{2}-\frac{16\sqrt{3}}{2})(\frac{18+17\sqrt{3}}{2}-\frac{36}{2})}[/tex]
[tex]A=\sqrt{\frac{18+17\sqrt{3}}{2}\cdot \frac{18-\sqrt{3}}{2} \cdot \frac{18+\sqrt{3}}{2}\cdot \frac{-18+17\sqrt{3}}{2}}[/tex]
[tex]A=\sqrt{(\frac{17\sqrt{3}+18}{2}\cdot \frac{17\sqrt{3}-18}{2})\cdot ( \frac{18-\sqrt{3}}{2} \cdot \frac{18+\sqrt{3}}{2})}\\ \\ A=\sqrt{\frac{(17\sqrt{3})^2-18^2}{4}\cdot \frac{18^2-(\sqrt{3})^2}{4}}[/tex]
[tex]A=\sqrt{\frac{867-324}{4}\cdot \frac{324-3}{4}}\\ \\ A=\sqrt{\frac{543}{4}\cdot \frac{321}{4}}\\ \\ A=\sqrt{\frac{174.303}{16}} \ aprox. \ 26,09 \ cm^2[/tex]
