Question

Se considera triunghiul PQS cu QP=6,sinS=2/3 și sinP=4/5.
Calculati lungimea laturii QS
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Answer 1

$\text{Folosim teorema sinusurilor:} \\ \\ \dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} = 2R = \dfrac{abc}{2S}\\ \\ \text{In }\triangle PQS \text{ avem:} \\\\\dfrac{p}{\sin P} = \dfrac{q}{\sin Q}= \dfrac{s}{\sin S} \Rightarrow \dfrac{QS}{\sin P} = \dfrac{PS}{\sin Q} = \dfrac{QP}{\sin S} \Rightarrow\\ \\ \Rightarrow \dfrac{QS}{\dfrac{4}{5}} = \dfrac{PS}{\sin Q}=\dfrac{6}{\dfrac{2}{3}} \\ \\ \\ \Rightarrow \dfrac{5QS}{4} = \dfrac{PS}{\sin Q} = \dfrac{6\cdot 3}{2}$

$\Rightarrow \dfrac{5QS}{4} = \dfrac{6\cdot 3}{2} \Rightarrow QS = 9\cdot \dfrac{4}{5} \Rightarrow QS = \dfrac{36}{5}$