Question

Aflați aria triunghiului ABC si sin(A),daca AB=17 cm,AC=25 cm si BC=28 cm.

Answer 1

$Formula~lui~Herron:\\\\ \boxed{A=\sqrt{p(p-a)(p-b)(p-c)}}\\\\unde:~p=semiperimetru=\frac{a+b+c}{2}\\\\si~a,~b,~c~lungimile~laturilor~triunghiului\\\\ \\\\ Inca~o~formula~pentru~aria~triunghiului\\\\ \boxed{A=\frac{a\cdot b \cdot sin(<a,~b)}{2}}\\\\ \\\\ \\\\ Intai~calculam~aria~triunghiului~cu~ajutorul~formulei~lui~Herron\\\\ p=\frac{17+25+28}{2}=\frac{70}{2}=35\\\\ \Rightarrow A=\sqrt{35(35-17)(35-25)(35-28)}=\\\\ =\sqrt{35\cdot 18 \cdot 10 \cdot 7}=\\\\ =\sqrt{5\cdot 7 \cdot 3^{2}\cdot 2 \cdot 2 \cdot 5 \cdot 7}=\\\\ =\sqrt{5^2 \cdot 7^2 \cdot 2^2 \cdot 3^2}=\\\\=5\cdot 7 \cdot 2 \cdot 3=\\\\ =210~(cm^{2})\\\\ \\\\ Din~\boxed{A=\frac{a\cdot b \cdot sin(<a,~b)}{2}} \Rightarrow A=\frac{AB\cdot AC \cdot sin(BAC)}{2}\Rightarrow \\\\ sin(BAC)=\frac{A\cdot 2}{AB \cdot AC}\\\\ sin(BAC)=\frac{210\cdot 2}{17 \cdot 25}=\frac{420}{425}=\frac{84}{85}$