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Rog-sadee

From Wikipedia, the free encyclopedia

Rog-sadee
Shallaggii naannoo roga sadiiShallaggii bal’ina roga saddeetiiGosoota Rogsadee

Rog-sadee jechuun danaa ji’oomeetirii sarara qajeelaa sadii fi guttiyyoo(vertex) sadii irraa hojjetame dha.

Rog-sadee
Odeeffannoo dabalataa : 1.1. ...
Dheerinni cinaachaa roga sadii a, b fi c yoo ta’u kofoon α, β fi γ dha.
  : 1.1. s = a + b + c {\displaystyle s=a+b+c} {\displaystyle s=a+b+c}
a sin ⁡ α = b sin ⁡ β = c sin ⁡ γ . {\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}.} {\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}.}
c 2   = a 2 + b 2 − 2 a b cos ⁡ ( γ ) {\displaystyle c^{2}\ =a^{2}+b^{2}-2ab\cos(\gamma )} {\displaystyle c^{2}\ =a^{2}+b^{2}-2ab\cos(\gamma )}
b 2   = a 2 + c 2 − 2 a c cos ⁡ ( β ) {\displaystyle b^{2}\ =a^{2}+c^{2}-2ac\cos(\beta )} {\displaystyle b^{2}\ =a^{2}+c^{2}-2ac\cos(\beta )}
a 2   = b 2 + c 2 − 2 b c cos ⁡ ( α ) {\displaystyle a^{2}\ =b^{2}+c^{2}-2bc\cos(\alpha )} {\displaystyle a^{2}\ =b^{2}+c^{2}-2bc\cos(\alpha )}
α = arccos ⁡ ( b 2 + c 2 − a 2 2 b c ) {\displaystyle \alpha =\arccos \left({\frac {b^{2}+c^{2}-a^{2}}{2bc}}\right)} {\displaystyle \alpha =\arccos \left({\frac {b^{2}+c^{2}-a^{2}}{2bc}}\right)}
β = arccos ⁡ ( a 2 + c 2 − b 2 2 a c ) {\displaystyle \beta =\arccos \left({\frac {a^{2}+c^{2}-b^{2}}{2ac}}\right)} {\displaystyle \beta =\arccos \left({\frac {a^{2}+c^{2}-b^{2}}{2ac}}\right)}
γ = arccos ⁡ ( a 2 + b 2 − c 2 2 a b ) {\displaystyle \gamma =\arccos \left({\frac {a^{2}+b^{2}-c^{2}}{2ab}}\right)} {\displaystyle \gamma =\arccos \left({\frac {a^{2}+b^{2}-c^{2}}{2ab}}\right)}
Cufi
Odeeffannoo dabalataa ...
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A r e a = 1 2 b h {\displaystyle \mathrm {Area} ={\frac {1}{2}}bh} {\displaystyle \mathrm {Area} ={\frac {1}{2}}bh}
  : 1.1. A r e a = 1 2 b h {\displaystyle \mathrm {Area} ={\frac {1}{2}}bh} {\displaystyle \mathrm {Area} ={\frac {1}{2}}bh} Bal’inni bal’ina, b cinaacha kamiyyuu roga saddeetii agarsiisa, h immoo olka’iinsa cinaacha kana irraa gara golee faallaatti perpendikulaariin kaafame agarsiisa.
A r e a = 1 2 a b sin ⁡ γ = 1 2 b c sin ⁡ α = 1 2 c a sin ⁡ β {\displaystyle Area={\frac {1}{2}}ab\sin \gamma ={\frac {1}{2}}bc\sin \alpha ={\frac {1}{2}}ca\sin \beta } {\displaystyle Area={\frac {1}{2}}ab\sin \gamma ={\frac {1}{2}}bc\sin \alpha ={\frac {1}{2}}ca\sin \beta }
A r e a = 1 2 a b sin ⁡ ( α + β ) = 1 2 b c sin ⁡ ( β + γ ) = 1 2 c a sin ⁡ ( γ + α ) . {\displaystyle Area={\frac {1}{2}}ab\sin(\alpha +\beta )={\frac {1}{2}}bc\sin(\beta +\gamma )={\frac {1}{2}}ca\sin(\gamma +\alpha ).} {\displaystyle Area={\frac {1}{2}}ab\sin(\alpha +\beta )={\frac {1}{2}}bc\sin(\beta +\gamma )={\frac {1}{2}}ca\sin(\gamma +\alpha ).}
A r e a = b 2 ( s i n α ) ( s i n ( α + β ) ) 2 s i n β {\displaystyle Area={\frac {b^{2}(sin\alpha )(sin(\alpha +\beta ))}{2sin\beta }}} {\displaystyle Area={\frac {b^{2}(sin\alpha )(sin(\alpha +\beta ))}{2sin\beta }}}
A r e a = a 2 2 ( c o t β + cot ⁡ γ ) = a 2 ( s i n β ) ( s i n γ ) 2 s i n ( β + γ ) {\displaystyle Area={\frac {a^{2}}{2(cot\beta +\cot \gamma )}}={\frac {a^{2}(sin\beta )(sin\gamma )}{2sin(\beta +\gamma )}}} {\displaystyle Area={\frac {a^{2}}{2(cot\beta +\cot \gamma )}}={\frac {a^{2}(sin\beta )(sin\gamma )}{2sin(\beta +\gamma )}}}
A r e a = s ( s − a ) ( s − b ) ( s − c ) {\displaystyle Area={\sqrt {s(s-a)(s-b)(s-c)}}} {\displaystyle Area={\sqrt {s(s-a)(s-b)(s-c)}}}

As s = a + b + c 2 {\displaystyle s={\frac {a+b+c}{2}}} {\displaystyle s={\frac {a+b+c}{2}}} Mee guddinni kofa 3 walakkaa naannoo ta’uu isaa haa hubannu

A r e a = 1 4 ( a 2 + b 2 + c 2 ) 2 − 2 ( a 4 + b 4 + c 4 ) {\displaystyle Area={\frac {1}{4}}{\sqrt {(a^{2}+b^{2}+c^{2})^{2}-2(a^{4}+b^{4}+c^{4})}}} {\displaystyle Area={\frac {1}{4}}{\sqrt {(a^{2}+b^{2}+c^{2})^{2}-2(a^{4}+b^{4}+c^{4})}}}
A r e a = 1 4 2 ( a 2 b 2 + a 2 c 2 + b 2 c 2 ) − ( a 4 + b 4 + c 4 ) {\displaystyle Area={\frac {1}{4}}{\sqrt {2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{4}+b^{4}+c^{4})}}} {\displaystyle Area={\frac {1}{4}}{\sqrt {2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{4}+b^{4}+c^{4})}}}
A r e a = 1 4 ( a + b − c ) ( a − b + c ) ( − a + b + c ) ( a + b + c ) . {\displaystyle Area={\frac {1}{4}}{\sqrt {(a+b-c)(a-b+c)(-a+b+c)(a+b+c)}}.} {\displaystyle Area={\frac {1}{4}}{\sqrt {(a+b-c)(a-b+c)(-a+b+c)(a+b+c)}}.}
Cufi
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    Rogsadee rogqixaa
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    Rog-sadee lamqixaa(isosceles)
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    Rog-sadee mitqixaa

Rogsadeen kofa sirri kan ballinni kofa saa 90 madaalkofa(digrii)

Rogsadeen kofqoree kan ballinni kofa isaa >90 madaalkofa(digrii)

Rogsadeen kofa jufunfula kan ballinni kofa saa <90 madaalkofa(digrii)

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Rogsadee kofasirri Rogsadee jufunfula (Obtuse) Rogsadee kofqoree(acute)
⏟ {\displaystyle \quad \underbrace {\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad } _{}} {\displaystyle \quad \underbrace {\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad } _{}}
Habalakaa (Obliikii)
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Shallaggii naannoo roga sadii

Shallaggii bal’ina roga saddeetii

Gosoota Rogsadee