Egy EFG háromszög szögei forditottan arányosak a 0,25, 0,1(6) és 0,125 számokkal. Számitsuk ki a szögek mértékét!
0 , 25 = 25 100 = 1 4 {\displaystyle 0,25={\frac {25}{100}}={\frac {1}{4}}} 0 , 1 ( 6 ) = 15 90 = 1 6 {\displaystyle 0,1(6)={\frac {15}{90}}={\frac {1}{6}}} 0 , 125 = 125 1000 = 1 8 {\displaystyle 0,125={\frac {125}{1000}}={\frac {1}{8}}} E ^ 1 1 4 = F ^ 1 1 6 = G ^ 1 1 8 {\displaystyle {\frac {\widehat {E}}{\frac {1}{\frac {1}{4}}}}={\frac {\widehat {F}}{\frac {1}{\frac {1}{6}}}}={\frac {\widehat {G}}{\frac {1}{\frac {1}{8}}}}} E ^ 4 = F ^ 6 = G ^ 8 {\displaystyle {\frac {\widehat {E}}{4}}={\frac {\widehat {F}}{6}}={\frac {\widehat {G}}{8}}} { 6 ⋅ E ^ = 4 ⋅ F ^ 8 ⋅ E ^ = 4 ⋅ G ^ 8 ⋅ F ^ = 6 ⋅ G ^ E ^ + F ^ + G ^ = 180 ∘ ⇒ { E ^ = 4 ⋅ F ^ : 6 8 ⋅ E ^ = 4 ⋅ G ^ 8 ⋅ F ^ = 6 ⋅ G ^ E ^ + F ^ + G ^ = 180 ∘ ⇒ { E ^ = 4 ⋅ F ^ : 6 8 ⋅ 4 ⋅ F ^ : 6 = 4 ⋅ G ^ 8 ⋅ F ^ = 6 ⋅ G ^ E ^ + F ^ + G ^ = 180 ∘ ⇒ { E ^ = 4 ⋅ F ^ : 6 G ^ = 8 ⋅ F ^ : 6 8 ⋅ F ^ = 6 ⋅ G ^ E ^ + F ^ + G ^ = 180 ∘ ⇒ {\displaystyle \left\{{\begin{array}{ll}6\cdot {\widehat {E}}=4\cdot {\widehat {F}}\\8\cdot {\widehat {E}}=4\cdot {\widehat {G}}\\8\cdot {\widehat {F}}=6\cdot {\widehat {G}}\\{\widehat {E}}+{\widehat {F}}+{\widehat {G}}=180^{\circ }\\\end{array}}\right.\Rightarrow \left\{{\begin{array}{ll}{\widehat {E}}=4\cdot {\widehat {F}}:6\\8\cdot {\widehat {E}}=4\cdot {\widehat {G}}\\8\cdot {\widehat {F}}=6\cdot {\widehat {G}}\\{\widehat {E}}+{\widehat {F}}+{\widehat {G}}=180^{\circ }\\\end{array}}\right.\Rightarrow \left\{{\begin{array}{ll}{\widehat {E}}=4\cdot {\widehat {F}}:6\\8\cdot 4\cdot {\widehat {F}}:6=4\cdot {\widehat {G}}\\8\cdot {\widehat {F}}=6\cdot {\widehat {G}}\\{\widehat {E}}+{\widehat {F}}+{\widehat {G}}=180^{\circ }\\\end{array}}\right.\Rightarrow \left\{{\begin{array}{ll}{\widehat {E}}=4\cdot {\widehat {F}}:6\\{\widehat {G}}=8\cdot {\widehat {F}}:6\\8\cdot {\widehat {F}}=6\cdot {\widehat {G}}\\{\widehat {E}}+{\widehat {F}}+{\widehat {G}}=180^{\circ }\\\end{array}}\right.\Rightarrow } ⇒ { E ^ = 4 ⋅ F ^ : 6 G ^ = 8 ⋅ F ^ : 6 F ^ = F ^ E ^ + F ^ + G ^ = 180 ∘ ⇒ { E ^ = 4 ⋅ F ^ : 6 G ^ = 8 ⋅ F ^ : 6 F ^ = F ^ 4 ⋅ F ^ : 6 + F ^ + 8 ⋅ F ^ : 6 = 180 ∘ ⇒ { E ^ = 4 ⋅ F ^ : 6 G ^ = 8 ⋅ F ^ : 6 F ^ = F ^ 4 ⋅ F ^ 6 + 6 ⋅ F ^ 6 + 8 ⋅ F ^ 6 = 180 ∘ ⇒ {\displaystyle \Rightarrow \left\{{\begin{array}{ll}{\widehat {E}}=4\cdot {\widehat {F}}:6\\{\widehat {G}}=8\cdot {\widehat {F}}:6\\{\widehat {F}}={\widehat {F}}\\{\widehat {E}}+{\widehat {F}}+{\widehat {G}}=180^{\circ }\\\end{array}}\right.\Rightarrow \left\{{\begin{array}{ll}{\widehat {E}}=4\cdot {\widehat {F}}:6\\{\widehat {G}}=8\cdot {\widehat {F}}:6\\{\widehat {F}}={\widehat {F}}\\4\cdot {\widehat {F}}:6+{\widehat {F}}+8\cdot {\widehat {F}}:6=180^{\circ }\\\end{array}}\right.\Rightarrow \left\{{\begin{array}{ll}{\widehat {E}}=4\cdot {\widehat {F}}:6\\{\widehat {G}}=8\cdot {\widehat {F}}:6\\{\widehat {F}}={\widehat {F}}\\{\frac {4\cdot {\widehat {F}}}{6}}+{\frac {6\cdot {\widehat {F}}}{6}}+{\frac {8\cdot {\widehat {F}}}{6}}=180^{\circ }\\\end{array}}\right.\Rightarrow } ⇒ { E ^ = 4 ⋅ F ^ : 6 G ^ = 8 ⋅ F ^ : 6 F ^ = F ^ 18 ⋅ F ^ 6 = 180 ∘ ⇒ { E ^ = 4 ⋅ F ^ : 6 G ^ = 8 ⋅ F ^ : 6 F ^ = F ^ 3 ⋅ F ^ = 180 ∘ ⇒ { E ^ = 4 ⋅ F ^ : 6 G ^ = 8 ⋅ F ^ : 6 F ^ = 60 ∘ ⇒ { E ^ = 4 ⋅ 60 ∘ : 6 G ^ = 8 ⋅ 60 ∘ : 6 F ^ = 60 ∘ ⇒ { E ^ = 4 ⋅ 10 ∘ G ^ = 8 ⋅ 10 ∘ F ^ = 60 ∘ ⇒ { E ^ = 40 ∘ F ^ = 60 ∘ G ^ = 80 ∘ {\displaystyle \Rightarrow \left\{{\begin{array}{ll}{\widehat {E}}=4\cdot {\widehat {F}}:6\\{\widehat {G}}=8\cdot {\widehat {F}}:6\\{\widehat {F}}={\widehat {F}}\\{\frac {18\cdot {\widehat {F}}}{6}}=180^{\circ }\\\end{array}}\right.\Rightarrow \left\{{\begin{array}{ll}{\widehat {E}}=4\cdot {\widehat {F}}:6\\{\widehat {G}}=8\cdot {\widehat {F}}:6\\{\widehat {F}}={\widehat {F}}\\3\cdot {\widehat {F}}=180^{\circ }\\\end{array}}\right.\Rightarrow \left\{{\begin{array}{ll}{\widehat {E}}=4\cdot {\widehat {F}}:6\\{\widehat {G}}=8\cdot {\widehat {F}}:6\\{\widehat {F}}=60^{\circ }\\\end{array}}\right.\Rightarrow \left\{{\begin{array}{ll}{\widehat {E}}=4\cdot 60^{\circ }:6\\{\widehat {G}}=8\cdot 60^{\circ }:6\\{\widehat {F}}=60^{\circ }\\\end{array}}\right.\Rightarrow \left\{{\begin{array}{ll}{\widehat {E}}=4\cdot 10^{\circ }\\{\widehat {G}}=8\cdot 10^{\circ }\\{\widehat {F}}=60^{\circ }\\\end{array}}\right.\Rightarrow \left\{{\begin{array}{ll}{\widehat {E}}=40^{\circ }\\{\widehat {F}}=60^{\circ }\\{\widehat {G}}=80^{\circ }\\\end{array}}\right.}
