Question

Aflati aria triunghiului ABC sin A ,daca AB=17 cm , AC=25 cm , BC=28 cm

Answer 1

Pentru arie folosim formula lui Heron:

$\it \mathcal{A} = \sqrt{p(p-a)(p-b)(p-c)} , \ unde \ p=\dfrac{a+b+c}{2}\ \ (semiperimetrul) \\ \\ p=\dfrac{17+25+28}{2} =\dfrac{70}{2} =35 \\ \\ p-a=35- 28=7 \\ \\ p-b=35-25=10 \\ \\ p-c=35-17=18$

$\it \mathcal{A} = \sqrt{35\cdot7\cdot10\cdot18} =\sqrt{7\cdot5\cdot7\cdot10\cdot18} =\sqrt{49\cdot900} =7\cdot30=210\ cm^2$

$\it \mathcal{A} = \dfrac{AB\cdot AC\cdot sinA}{2} \Rightarrow 210=\dfrac{17\cdot25\cdot sinA}{2} \Rightarrow 210\cdot2=17\cdot25\cdot sinA\Rightarrow \\ \\ \\ \Rightarrow sinA=\dfrac{210\cdot2}{17\cdot25} = \dfrac{84}{85}$