Question

fie triunghiul ABC aflati aria triunghiului  daca AB=18cm BC=9√3cm  AC=8√3cm
ajutatima va rog

Answer 1

Atunci cand problema iti cere aria triunghiului si ti se dau toate laturile, poti sa:

•  verifici daca e triunghi dreptunghic - folosesti formula A=c1*c2/2 sau echilateral - folosesti formula A=
•  calculezi aria direct cu formula lui Herron

Formula lui Herron:

$\boxed{A=\sqrt{p(p-a)(p-b)(p-c)}}$

• a, b  si c sunt laturile triunghiului
• p - este semiperimetrul triunghiului, p=(a+b+c)/2

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

$A=\sqrt{\frac{18+17\sqrt{3}}{2}(\frac{18+17\sqrt{3}}{2}-9\sqrt{3})(\frac{18+17\sqrt{3}}{2}-8\sqrt{3})(\frac{18+17\sqrt{3}}{2}-18)}$

$A=\sqrt{\frac{18+17\sqrt{3}}{2}(\frac{18+17\sqrt{3}}{2}-\frac{18\sqrt{3}}{2})(\frac{18+17\sqrt{3}}{2}-\frac{16\sqrt{3}}{2})(\frac{18+17\sqrt{3}}{2}-\frac{36}{2})}$

$A=\sqrt{\frac{18+17\sqrt{3}}{2}\cdot \frac{18-\sqrt{3}}{2} \cdot \frac{18+\sqrt{3}}{2}\cdot \frac{-18+17\sqrt{3}}{2}}$

$A=\sqrt{(\frac{17\sqrt{3}+18}{2}\cdot \frac{17\sqrt{3}-18}{2})\cdot ( \frac{18-\sqrt{3}}{2} \cdot \frac{18+\sqrt{3}}{2})}\\ \\ A=\sqrt{\frac{(17\sqrt{3})^2-18^2}{4}\cdot \frac{18^2-(\sqrt{3})^2}{4}}$

$A=\sqrt{\frac{867-324}{4}\cdot \frac{324-3}{4}}\\ \\ A=\sqrt{\frac{543}{4}\cdot \frac{321}{4}}\\ \\ A=\sqrt{\frac{174.303}{16}} \ aprox. \ 26,09 \ cm^2$