Question

In triunghiul dreptunghic ABC, m(<ACB)=90⁰, Ac=a, BC=b, A-aria ΔABC, Aflati masurile  unghiurilor BAC si ABC, daca se cunoaste ca (a+b)²=8A

Answer 1

$\displaystyle A=\frac{(a+b)^2}{8}$

$\displaystyle A=\frac{ab}{2}.$

$\displaystyle \Rightarrow\frac{(a+b)^2}{8}=\frac{ab}{2}.$

Notam $t=\text{tg}(BAC).$

Atunci  $t=\frac{b}{a}\Rightarrow b=at$

Inlocuim: $\displaystyle \frac{(a+at)^2}{8}=\frac{a\cdot at}{2}$

$\displaystyle \frac{a^2(1+t)^2}{8}=\frac{a^2t}{2}$

$\displaystyle \frac{(1+t)^2}{4}=t$

$(1+t)^2=4t \\ \\ t^2+2t+1=4t \\ \\ t^2-2t+1=0 \\ \\ (t-1)^2=0\Rightarrow t=1.$

Deci m(BAC)=45 grade

Prin urmare si m(ABC)=45.